######  Theorem 0.8.2.

Let $\mathsf{G}$ be a nilpotent simply connected Lie group admitting flat
orbits. Then $(U_{\boldsymbol{B}})_{\boldsymbol{B}\in\mathfrak{JH}}$ is an
open cover of $\Gamma$.

To prove Theorem 0.8.2, we shall first need a lemma.

######  Lemma 0.8.3.

Let $\mathsf{G}$ be a nilpotent simply connected Lie group with a flat orbit
through $\xi\in\mathfrak{z}^{*}\subseteq\mathfrak{g}^{*}$. Let
$\Gamma_{\mathsf{G}}$ denote the set of flat orbits for $\mathsf{G}$. Fix an
$X\in\ker(\xi)\cap\mathfrak{z}$. The following holds the Lie algebra
$\bar{\mathfrak{g}}:=\mathfrak{g}/\mathbb{R}X$:

  1. i)

$\bar{\mathfrak{g}}$ admits a flat orbit through the element
$\bar{\xi}\in(\bar{\mathfrak{g}})^{*}$ induced by $\xi$.

  2. ii)

The centre $\bar{\mathfrak{z}}$ of $\bar{\mathfrak{g}}$ is given by
$\bar{\mathfrak{z}}=\mathfrak{z}/\mathbb{R}X$.

  3. iii)

The set $\Gamma_{\bar{\mathsf{G}}}$ of flat orbits for $\bar{\mathsf{G}}$ can
be identified with a hypersurface in $\Gamma_{\mathsf{G}}$ via the injection
$\bar{\mathfrak{z}}^{*}\hookrightarrow\mathfrak{z}^{*}$ dual to the projection
from item ii).

###### Proof.

Since $\xi$ lies in a flat orbit,

$\mathfrak{z}/\mathbb{R}X=\mathsf{stab}(\xi)/\mathbb{R}X=\mathsf{stab}(\bar{\xi}).$

We conclude from the series of inclusions
$\bar{\mathfrak{z}}\subseteq\mathsf{stab}(\bar{\xi})=\mathfrak{z}/\mathbb{R}X\subseteq\bar{\mathfrak{z}}$
that $\mathfrak{z}/\mathbb{R}X=\bar{\mathfrak{z}}$ and that $\bar{\xi}$ lies
in a flat orbit. The lemma follows. ∎

###### Proof of Theorem 0.8.2.

We prove the theorem by induction on the dimension of the centre of
$\mathsf{G}$. If $\mathsf{G}$ has one-dimensional centre and flat orbits, the
statement holds because $\Gamma=\mathfrak{z}^{*}\setminus\\{0\\}$ and
$\Gamma=U_{\boldsymbol{B}}$ for any $\boldsymbol{B}\in\mathfrak{JH}$. Assume
that the statement of the theorem holds true for nilpotent simply connected
Lie groups admitting flat orbits with centre of dimension $\leq m-1$. Consider
a nilpotent simply connected Lie group $\mathsf{G}$ admitting flat orbits and
whose centre is of dimension $m$. Take a $\xi\in\Gamma_{\mathsf{G}}$ and an
$X\in\ker(\xi)\cap\mathfrak{z}$. We follow the notation of Lemma 0.8.3.

The nilpotent simply connected Lie group $\bar{\mathsf{G}}$ that
$\bar{\mathfrak{g}}$ integrates to satisfy the statement of the theorem by the
induction assumption, because $\bar{\mathfrak{z}}=\mathfrak{z}/\mathbb{R}X$
has dimension $m-1$. Therefore, there is a Jordan-Hölder basis
$\boldsymbol{B}_{1}=(\bar{X}_{2},\ldots\bar{X}_{n})$ for $\bar{\mathfrak{g}}$
such that $\bar{\xi}\in
U_{\boldsymbol{B}_{1}}\subseteq\Gamma_{\bar{\mathsf{G}}}$. We can lift
$(\bar{X}_{2},\ldots\bar{X}_{n})$ to $X_{2},\ldots,X_{n}\in\mathfrak{g}$ and,
up to permutation, consider the Jordan-Hölder basis
$\boldsymbol{B}=(X_{1},X_{2},\ldots,X_{n})$ for $\mathfrak{g}$ where
$X_{1}=X$.

The induction step is complete upon proving that
$\Gamma_{J_{\xi}}^{\boldsymbol{B}}$ contains an open subset, in which case
$\Gamma_{J_{\xi}}^{\boldsymbol{B}}=U_{\boldsymbol{B}}$ and $\xi\in
U_{\boldsymbol{B}}$ since $\xi\in\Gamma_{J_{\xi}}^{\boldsymbol{B}}$ by
definition. Take an $\eta\in\mathfrak{z}^{*}$ such that $\eta(X)=1$ and
$\eta(X_{j})=0$ for $j>1$. By Lemma 0.8.3.iii), it suffices to prove that
$J_{\xi^{\prime}+t\eta}^{\boldsymbol{B}}$ is independent of sufficiently small
$t$ for $\xi^{\prime}\in
U_{\boldsymbol{B}_{1}}\subseteq\Gamma_{\bar{\mathsf{G}}}$ in a neighbourhood
of $\bar{\xi}$. Consider the family of Jordan-Hölder bases
$\boldsymbol{B}(t)=(X_{1}-tX_{2},X_{2},\ldots,X_{n})$. By definition, we have
that
$J_{\xi^{\prime}+t\eta}^{\boldsymbol{B}}=J_{\xi^{\prime}}^{\boldsymbol{B}(t)}$
and since the variation in $t$ only can alter jump indices in the first step,
we have that $J_{\xi^{\prime}}^{\boldsymbol{B}(t)}$ is independently of
sufficiently small $t$. We conclude that
$J_{\xi^{\prime}+t\eta}^{\boldsymbol{B}}$ is independent of sufficiently small
$t$, and $\Gamma_{J_{\xi}}^{\boldsymbol{B}}$ contains an open neighbourhood
around $\xi$. ∎

######  Remark 0.8.4.

Consider the statement:

_Assume that $\mathsf{G}$ has orbits of dimension at most $2d$, and consider
the space of orbits of maximal dimension
$\tilde{\Gamma}:=\\{\mathcal{O}\in\hat{\mathsf{G}}:\dim(\mathcal{O})=2d\\}$.
Then $(U_{\boldsymbol{B}})_{\boldsymbol{B}\in\mathfrak{JH}}$ is an open cover
of $\tilde{\Gamma}$. _

Theorem 0.8.2 shows that the above statement holds in the presence of flat
orbits – in this case $\Gamma=\tilde{\Gamma}$. Generally we have that
$\cup_{\boldsymbol{B}\in\mathfrak{JH}}U_{\boldsymbol{B}}\subseteq\tilde{\Gamma}$
is Zariski open (since each $U_{\boldsymbol{B}}$ is). However, the statement
above fails in general as seen from the four dimensional Lie algebra spanned
by $X_{1},X_{2},X_{3},X_{4}$ with brackets $[X_{4},X_{3}]=X_{2}$ and
$[X_{4},X_{2}]=X_{1}$ and all other brackets zero. For this Lie algebra we can
identify $\tilde{\Gamma}=\\{(\xi_{1},\xi_{2}):(\xi_{1},\xi_{2})\neq 0\\}$ but
$U_{\boldsymbol{B}}=\\{(\xi_{1},\xi_{2}):\xi_{1}\neq 0\\}$ for any Jordan-
Hölder basis $\boldsymbol{B}\in\mathfrak{JH}$. This example can be found in
[128, Remark 1, p. 274].

To simplify notation, we shall drop the notation involving jump indices, as we
henceforth only use the minimal jump index that corresponds to the top strata,
and instead indicate the basis. For instance, $U_{\boldsymbol{B}}$ denotes the
top fine strata in the stratification defined from the basis $\boldsymbol{B}$,
$F_{V,\boldsymbol{B}}\to U_{\boldsymbol{B}}$ denotes the bundle of Vergne
polarizations, and $\mathcal{H}_{V,\boldsymbol{B}}\to U_{\boldsymbol{B}}$ the
associated Hilbert space bundle of $\mathsf{G}$-representations over
$U_{\boldsymbol{B}}\subseteq\Gamma\subseteq\hat{\mathsf{G}}$.

For any $\boldsymbol{B}_{1},\ldots,\boldsymbol{B}_{k}\in\mathfrak{JH}$, we
write

$\displaystyle I_{\mathsf{G},\boldsymbol{B}_{1},\ldots,\boldsymbol{B}_{k}}:$
$\displaystyle=C_{0}(U_{\boldsymbol{B}_{1}}\cap\cdots
U_{\boldsymbol{B}_{k}})I_{\mathsf{G}}=$ $\displaystyle=\\{a\in
C^{*}(\mathsf{G}):\pi(a)=0\;\forall\pi\in\hat{\mathsf{G}}\setminus
U_{\boldsymbol{B}_{1}}\cap\cdots U_{\boldsymbol{B}_{k}}\\}.$

For each $\boldsymbol{B}\in\mathfrak{JH}$, Proposition 0.5.7 and 0.5.9
produces canonical trivializations

$\pi_{\boldsymbol{B}}:I_{\mathsf{G},\boldsymbol{B}}\to
C_{0}(U_{\boldsymbol{B}},\mathbb{K}(\mathcal{H}_{V,\boldsymbol{B}})),$

where $d=\mathrm{codim}(\mathfrak{z})/2$. For any
$\boldsymbol{B},\boldsymbol{B}^{\prime}\in\mathfrak{JH}$, the corrected Lion
intertwiner from Proposition 0.4.5 (see also Lemma 0.4.2) defines a fibrewise
unitary intertwiner

$\mathfrak{L}_{\boldsymbol{B},\boldsymbol{B}^{\prime}}:=\mathfrak{L}_{F_{V,\boldsymbol{B}^{\prime}},F_{V,\boldsymbol{B}}}:\mathcal{H}_{V,\boldsymbol{B}^{\prime}}|_{U_{\boldsymbol{B}}\cap
U_{\boldsymbol{B}^{\prime}}}\to\mathcal{H}_{V,\boldsymbol{B}}|_{U_{\boldsymbol{B}}\cap
U_{\boldsymbol{B}^{\prime}}}.$

By possibly shrinking the open sets in the cover, we can also ensure
continuity. In other words,

$\mathsf{Ad}(\mathfrak{L}_{\boldsymbol{B}^{\prime},\boldsymbol{B}}):C_{0}(U_{\boldsymbol{B}}\cap
U_{\boldsymbol{B}^{\prime}},\mathbb{K}(\mathcal{H}_{V,\boldsymbol{B}}|_{U_{\boldsymbol{B}}\cap
U_{\boldsymbol{B}^{\prime}}}))\to C_{0}(U_{\boldsymbol{B}}\cap
U_{\boldsymbol{B}^{\prime}},\mathbb{K}(\mathcal{H}_{V,\boldsymbol{B}^{\prime}}|_{U_{\boldsymbol{B}}\cap
U_{\boldsymbol{B}^{\prime}}})),$

is a unitary isomorphism fitting into the commuting diagram of
$*$-isomorphisms

${C_{0}(U_{\boldsymbol{B}}\cap
U_{\boldsymbol{B}^{\prime}},\mathbb{K}(\mathcal{H}_{V,\boldsymbol{B}}|_{U_{\boldsymbol{B}}\cap
U_{\boldsymbol{B}^{\prime}}}))}$${C_{0}(U_{\boldsymbol{B}}\cap
U_{\boldsymbol{B}^{\prime}},\mathbb{K}(\mathcal{H}_{V,\boldsymbol{B}^{\prime}}|_{U_{\boldsymbol{B}}\cap
U_{\boldsymbol{B}^{\prime}}}))}$${I_{G,\boldsymbol{B},\boldsymbol{B}^{\prime}}}$$\scriptstyle{\mathsf{Ad}(\mathfrak{L}_{\boldsymbol{B}^{\prime},\boldsymbol{B}})}$$\scriptstyle{\pi_{\boldsymbol{B}}}$$\scriptstyle{\pi_{\boldsymbol{B}^{\prime}}}$

We conclude the following from Proposition 0.4.5.

######  Proposition 0.8.5.

Let $\mathsf{G}$ be a simply connected nilpotent Lie group admitting flat
orbits. Then the collection
$(\mathcal{H}_{V,\boldsymbol{B}})_{\boldsymbol{B}\in\mathfrak{JH}}$ glues
together as bundles of Hilbert spaces with a fibrewise unitary
$\mathsf{G}$-action to a bundle of flat representations

$\mathcal{H}_{\Gamma}\to\Gamma,$

via the corrected Lion intertwiners
$(\mathfrak{L}_{\boldsymbol{B},\boldsymbol{B}^{\prime}})_{\boldsymbol{B},\boldsymbol{B}^{\prime}\in\mathfrak{JH}}$
from Proposition 0.4.5. In particular, the fibrewise unitary
$\mathsf{G}$-action induces a $C_{0}(\Gamma)$-linear $*$-isomorphism

$\pi_{\musFlat}:I_{\mathsf{G}}\to
C_{0}(\Gamma,\mathbb{K}(\mathcal{H}_{\Gamma})).$

The existence of a bundle of flat orbit representations also follow from a
deformation argument below in Theorem 0.14.6.

By the universal property of the group $C^{*}$-algebra, any group automorphism
$\varphi\in\mathrm{Aut}(G)$ induces an automorphism $C^{*}(\mathsf{G})$ that
we by an abuse of notation also denote by $\varphi$. Since the associated
group homomorphism
$\mathrm{Aut}(\mathsf{G})\to\mathrm{Aut}(C^{*}(\mathsf{G}))$ is injective this
abuse of notation should not be a cause of confusion. By Proposition 0.5.8 and
0.7.1, any automorphism $\varphi\in\mathrm{Aut}(\mathsf{G})$ preserves the
ideal $I_{\mathsf{G}}\subseteq C^{*}(\mathsf{G})$ of flat orbits. We
abstractly know that $I_{\mathsf{G}}$ is a continuous trace algebra over
$\Gamma$ with fibre $\mathbb{K}(L^{2}(\mathbb{R}^{d}))$, where
$d=\mathrm{codim}(\mathfrak{z})/2$, and $I_{\mathsf{G}}$ trivializes by
Proposition 0.8.5. We now describe the action of automorphisms on
$I_{\mathsf{G}}$ and their lifts to the bundle of flat orbit representations.

For a Jordan-Hölder basis
$\boldsymbol{B}=(X_{1},\ldots,X_{n})\in\mathfrak{JH}$ of $\mathfrak{g}$, with
$\mathfrak{g}_{\dim\mathfrak{z}}=\mathfrak{z}$ as in Section 0.5 and 0.6,
recall the construction of $F_{V,\boldsymbol{B}}\to U_{\boldsymbol{B}}$ from
Proposition 0.5.2, where we write $F_{V,\boldsymbol{B}}$ and
$U_{\boldsymbol{B}}$ for the top strata in the basis $\boldsymbol{B}$ (cf.
Section 0.6). For a $\varphi\in\mathrm{Aut}(\mathsf{G})$, we write
$\varphi(\boldsymbol{B})\in\mathfrak{JH}$ for the Jordan-Hölder basis

$\varphi(\boldsymbol{B}):=(\varphi^{-1}X_{1},\ldots,\varphi^{-1}X_{n}).$

######  Proposition 0.8.6.

Let $\varphi,\varphi^{\prime}$ be automorphisms of a simply connected
nilpotent Lie group $\mathsf{G}$ admitting flat orbits. Take
$\boldsymbol{B},\boldsymbol{B}^{\prime}\in\mathfrak{JH}$.

  1. a)

The mapping

$\varphi^{*}\times\varphi^{-1}:\mathfrak{g}^{*}\times\mathfrak{g}\to\mathfrak{g}^{*}\times\mathfrak{g}$

restricts to a homeomorphism
$\varphi^{*}\times\varphi^{-1}:F_{V,\boldsymbol{B}}\to
F_{V,\varphi(\boldsymbol{B})}$ that fits into a commuting diagram

${F_{V,\boldsymbol{B}}}$${F_{V,\varphi(\boldsymbol{B})}}$${\
U_{\boldsymbol{B}}}$${U_{\varphi(\boldsymbol{B})}.}$$\scriptstyle{\varphi^{*}\times\varphi^{-1}}$$\scriptstyle{\varphi^{*}}$

  2. b)

The map $(\xi,f)\mapsto(\varphi^{*}\xi,f\circ\varphi^{*})$ defines a fibrewise
unitary morphism

$t_{\boldsymbol{B}}(\varphi):\mathcal{H}_{V,\boldsymbol{B}}\to\mathcal{H}_{V,\varphi(\boldsymbol{B})},$

such that:

     * •

The following diagram of Hilbert space bundles commutes

${\mathcal{H}_{V,\boldsymbol{B}}}$${\mathcal{H}_{V,\varphi(\boldsymbol{B})}}$${\
U_{\boldsymbol{B}}}$${U_{\varphi(\boldsymbol{B})},}$$\scriptstyle{t_{\boldsymbol{B}}(\varphi)}$$\scriptstyle{\varphi^{*}}$

and $t_{\boldsymbol{B}}(\varphi)$ is equivariant for the ordinary
$\mathsf{G}$-action on $\mathcal{H}_{V,\boldsymbol{B}}$ and the
$\mathsf{G}$-action on $\mathcal{H}_{V,\varphi(\boldsymbol{B})}$ pulled back
along $\varphi$.

     * •

It holds that

$t_{\varphi(\boldsymbol{B})}(\varphi^{\prime})t_{\boldsymbol{B}}(\varphi)=t_{\boldsymbol{B}}(\varphi\varphi^{\prime}),$

as morphisms of Hilbert space bundles
$\mathcal{H}_{V,\boldsymbol{B}}\to\mathcal{H}_{V,\varphi\varphi^{\prime}(\boldsymbol{B})}$.

     * •

The following diagram of Hilbert space bundles commutes

${\mathcal{H}_{V,\boldsymbol{B}}|_{U_{\boldsymbol{B}}\cap
U_{\boldsymbol{B}^{\prime}}}}$${\mathcal{H}_{V,\varphi(\boldsymbol{B})}|_{U_{\varphi(\boldsymbol{B})}\cap
U_{\varphi(\boldsymbol{B}^{\prime})}}}$${\
\mathcal{H}_{V,\boldsymbol{B}^{\prime}}|_{U_{\boldsymbol{B}}\cap
U_{\boldsymbol{B}^{\prime}}}}$${\mathcal{H}_{V,\varphi(\boldsymbol{B}^{\prime})}|_{U_{\varphi(\boldsymbol{B})}\cap
U_{\varphi(\boldsymbol{B}^{\prime})}}}$$\scriptstyle{\mathfrak{L}_{\boldsymbol{B}^{\prime},\boldsymbol{B}}^{(0)}}$$\scriptstyle{t_{\boldsymbol{B}}(\varphi)}$$\scriptstyle{\mathfrak{L}_{\varphi(\boldsymbol{B}^{\prime}),\varphi(\boldsymbol{B})}^{(0)}}$$\scriptstyle{t_{\boldsymbol{B}^{\prime}}(\varphi)}$

  3. c)

The automorphism $\varphi\in\mathrm{Aut}(I_{\mathsf{G}})$ restricts to a
$*$-isomorphism $\varphi|:I_{\mathsf{G},\boldsymbol{B}}\to
I_{\mathsf{G},\varphi(\boldsymbol{B})}$ that together with
$t_{\boldsymbol{B}}(\varphi)$ from the previous item fit into the commuting
diagram of $*$-isomorphisms

${I_{\mathsf{G},\boldsymbol{B}}}$${I_{\mathsf{G},\varphi(\boldsymbol{B})}}$${C_{0}(U_{\boldsymbol{B}},\mathbb{K}(\mathcal{H}_{V,\boldsymbol{B}}))}$${C_{0}(U_{\varphi(\boldsymbol{B)}},\mathbb{K}(\mathcal{H}_{V,\varphi(\boldsymbol{B})})).}$$\scriptstyle{\pi_{\boldsymbol{B}}}$$\scriptstyle{\varphi|}$$\scriptstyle{\pi_{\varphi(\boldsymbol{B})}}$$\scriptstyle{\mathrm{Ad}(t_{\boldsymbol{B}}(\varphi))}$

###### Proof.

Note that $\mathfrak{h}$ is a real algebraic polarization of
$\xi\in\mathfrak{g}^{*}$ if and only if $\varphi^{-1}\mathfrak{h}$ is a real
algebraic polarization of $\varphi^{*}\xi\in\mathfrak{g}^{*}$. We also note
that since
$\varphi^{*}(\Xi_{\varepsilon,\boldsymbol{B}})=\Xi_{\varepsilon,\varphi(\boldsymbol{B})}$
for any $\varepsilon$, and $\varphi^{*}$ is a homeomorphism on $\Gamma$, we
have that $\varphi^{*}(U_{\boldsymbol{B}})=U_{\varphi(\boldsymbol{B})}$. To
prove item a), we simply note that for $\xi\in U_{\varphi(\boldsymbol{B})}$,

$\varphi^{-1}(\mathfrak{h}_{V,\boldsymbol{B}}(\varphi^{-1*}\xi))=\mathfrak{h}_{V,\varphi(\boldsymbol{B})}(\xi),$

which implies that

$\displaystyle(\varphi^{*}\times\varphi^{-1})F_{V,\boldsymbol{B}}$
$\displaystyle=\\{(\varphi^{*}\xi,\varphi^{-1}X):\,\xi\in
U_{\boldsymbol{B}},X\in\mathfrak{h}_{V,\boldsymbol{B}}(\xi)\\}=$
$\displaystyle=\\{(\xi,X):\,\xi\in
U_{\varphi(\boldsymbol{B})},X\in\varphi^{-1}(\mathfrak{h}_{V,\boldsymbol{B}}(\varphi^{-1*}\xi)\\}=F_{V,\varphi(\boldsymbol{B})}.$

Item b) and c) are proven by similar, lengthier but straightforward,
computations and we omit their proofs. ∎

To describe how to lift an automorphism to the bundle $\mathcal{H}_{\Gamma}$
of flat orbit representations constructed in Proposition 0.8.5, we follow
[131]. We let
$(u_{\boldsymbol{B},\boldsymbol{B}^{\prime}})_{\boldsymbol{B},\boldsymbol{B}^{\prime}\in\mathfrak{JH}}\in\check{C}^{1}((U_{\boldsymbol{B}})_{\boldsymbol{B}\in\mathfrak{JH}},U(1))$
denote the cocycle defined from

$u_{\boldsymbol{B},\boldsymbol{B}^{\prime}}:=\mathrm{e}^{\frac{\pi
i}{4}\eta(F_{V,\boldsymbol{B}^{\prime}},F_{V,\boldsymbol{B}})}.$

Here we have for notational simplicity disregarded the fact that the covering
needs to be refined to ensure continuity. We define the map

$\displaystyle\zeta_{0}:\mathrm{Aut}(\mathsf{G})\to$
$\displaystyle\check{C}^{1}(((U_{\boldsymbol{B}})_{\boldsymbol{B}\in\mathfrak{JH}},U(1))),$
(14)
$\displaystyle\varphi\mapsto(\varphi^{*}(u_{\boldsymbol{B},\boldsymbol{B}^{\prime}})u_{\varphi(\boldsymbol{B}),\varphi(\boldsymbol{B}^{\prime})}^{-1})_{\boldsymbol{B},\boldsymbol{B}^{\prime}\in\mathfrak{JH}}.$

######  Proposition 0.8.7.

Let $\mathsf{G}$ be a simply connected nilpotent Lie group admitting flat
orbits. Then the mapping $\zeta_{0}$ satisfies

  1. (1)

The range of $\zeta_{0}$ is contained in the Cech cocycles, i.e. $\zeta_{0}$
induces a mapping

$\zeta:\mathrm{Aut}(\mathsf{G})\to\check{H}^{1}(\Gamma,U(1)).$

  2. (2)

$\zeta_{0}\in
Z^{1}(\mathrm{Aut}(\mathsf{G}),\check{Z}^{1}((U_{\boldsymbol{B}})_{\boldsymbol{B}\in\mathfrak{JH}},U(1)))$
is a group cocycle for the induced $\mathrm{Aut}(\mathsf{G})$-action on
$\check{Z}^{1}((U_{\boldsymbol{B}})_{\boldsymbol{B}\in\mathfrak{JH}},U(1))$,
and induces a cohomology class

$[\zeta_{0}]\in
H^{1}(\mathrm{Aut}(\mathsf{G}),\check{Z}^{1}((U_{\boldsymbol{B}})_{\boldsymbol{B}\in\mathfrak{JH}},U(1))),$

and $\zeta\in Z^{1}(\mathrm{Aut}(\mathsf{G}),\check{H}^{1}(\Gamma,U(1)))$
induces a cohomology class

$[\zeta]\in H^{1}(\mathrm{Aut}(\mathsf{G}),\check{H}^{1}(\Gamma,U(1))).$

  3. (3)

The class $[\zeta]\in
H^{1}(\mathrm{Aut}(\mathsf{G}),\check{H}^{1}(\Gamma,U(1)))$ does not depend on
the choice of
$(u_{\boldsymbol{B},\boldsymbol{B}^{\prime}})_{\boldsymbol{B},\boldsymbol{B}^{\prime}\in\mathfrak{JH}}\in\check{C}^{1}((U_{\boldsymbol{B}})_{\boldsymbol{B}\in\mathfrak{JH}},U(1))$
as long as the cochain
$(u_{\boldsymbol{B},\boldsymbol{B}^{\prime}})_{\boldsymbol{B},\boldsymbol{B}^{\prime}}$
solves

$u_{\boldsymbol{B},\boldsymbol{B}^{\prime}}u_{\boldsymbol{B}^{\prime},\boldsymbol{B}^{\prime\prime}}u_{\boldsymbol{B}^{\prime\prime},\boldsymbol{B}}=\mathrm{e}^{\frac{\pi
i}{4}\mathrm{Mas}(F_{V,\boldsymbol{B}},F_{V,\boldsymbol{B}^{\prime}},F_{V,\boldsymbol{B}^{\prime\prime}})}.$
(15)

Moreover, the action of $\varphi\in\mathrm{Aut}(\mathsf{G})$ on $\Gamma$ lifts
to a bundle morphism on $\mathcal{H}_{\Gamma}$ inducing $\varphi$ on
$I_{\mathsf{G}}$ if and only if
$\zeta(\varphi)=0\in\check{H}^{1}(\Gamma,U(1))$. Additionally, for a line
bundle $L\to\Gamma$, $\varphi$ lifts to a bundle morphism on
$\mathcal{H}_{\Gamma}\otimes L$ inducing $\varphi$ on $I_{\mathsf{G}}$ if and
only if
$\zeta(\varphi)=\varphi^{*}(c_{1}(L))c_{1}(L^{-1})\in\check{H}^{1}(\Gamma,U(1))$.

###### Proof.

The coboundary of $\zeta_{0}(\varphi)$ is the Cech cocycle given by

$\displaystyle\varphi^{*}(u_{\boldsymbol{B},\boldsymbol{B}^{\prime}})$
$\displaystyle
u_{\varphi(\boldsymbol{B}),\varphi(\boldsymbol{B}^{\prime})}^{-1}\varphi^{*}(u_{\boldsymbol{B^{\prime}},\boldsymbol{B}^{\prime\prime}})u_{\varphi(\boldsymbol{B}^{\prime}),\varphi(\boldsymbol{B}^{\prime\prime})}^{-1}\varphi^{*}(u_{\boldsymbol{B}^{\prime\prime},\boldsymbol{B}})u_{\varphi(\boldsymbol{B}^{\prime\prime}),\varphi(\boldsymbol{B})}^{-1}=$
$\displaystyle=t_{\boldsymbol{B}}(\varphi)(\mathfrak{L}_{\boldsymbol{B},\boldsymbol{B}^{\prime}}\mathfrak{L}_{\boldsymbol{B}^{\prime},\boldsymbol{B}^{\prime\prime}}\mathfrak{L}_{\boldsymbol{B}^{\prime\prime},\boldsymbol{B}})t_{\boldsymbol{B}}(\varphi)^{*}\mathfrak{L}_{\varphi(\boldsymbol{B}),\varphi(\boldsymbol{B}^{\prime})}\mathfrak{L}_{\varphi(\boldsymbol{B}^{\prime}),\varphi(\boldsymbol{B}^{\prime\prime})}\mathfrak{L}_{\varphi(\boldsymbol{B}^{\prime\prime}),\varphi(\boldsymbol{B})}=1,$

by Proposition 0.8.6. Item 1 follows. To prove item 2, we note that the group
coboundary $\delta\zeta_{0}(\varphi,\varphi^{\prime})$ is given by

$\displaystyle\delta$
$\displaystyle\zeta_{0}(\varphi,\varphi^{\prime})=\varphi^{*}\zeta_{0}(\varphi^{\prime})\zeta_{0}(\varphi\varphi^{\prime})^{-1}\zeta_{0}(\varphi)=$
$\displaystyle=((\varphi\varphi^{\prime})^{*}(u_{\boldsymbol{B},\boldsymbol{B}^{\prime}})\varphi^{*}u_{\boldsymbol{B},\boldsymbol{B}^{\prime}}^{-1}(\varphi\varphi^{\prime})^{*}(u_{\boldsymbol{B},\boldsymbol{B}^{\prime}})^{-1}u_{\boldsymbol{B},\boldsymbol{B}^{\prime}}\varphi^{*}(u_{\boldsymbol{B},\boldsymbol{B}^{\prime}})u_{\boldsymbol{B},\boldsymbol{B}^{\prime}}^{-1})_{\boldsymbol{B},\boldsymbol{B}^{\prime}\in\mathfrak{JH}}=1.$

To prove item 3), we note that if
$(u^{\prime}_{\boldsymbol{B},\boldsymbol{B}^{\prime}})_{\boldsymbol{B},\boldsymbol{B}^{\prime}\in\mathfrak{JH}}\in\check{C}^{1}((U_{\boldsymbol{B}})_{\boldsymbol{B}\in\mathfrak{JH}},U(1))$
is another choice of $U(1)$-cochain solving (15), then
$(u^{\prime}_{\boldsymbol{B},\boldsymbol{B}^{\prime}}u^{-1}_{\boldsymbol{B},\boldsymbol{B}^{\prime}})_{\boldsymbol{B},\boldsymbol{B}^{\prime}\in\mathfrak{JH}}\in\check{Z}^{1}((U_{\boldsymbol{B}})_{\boldsymbol{B}\in\mathfrak{JH}},U(1))$.
Denote the associated cohomology class by $x\in\check{H}^{1}(\Gamma,U(1))$ and
let $\zeta^{\prime}\in
Z^{1}(\mathrm{Aut}(\mathsf{G}),\check{H}^{1}(\Gamma,U(1)))$ denote the cocycle
constructed from
$(u^{\prime}_{\boldsymbol{B},\boldsymbol{B}^{\prime}})_{\boldsymbol{B},\boldsymbol{B}^{\prime}\in\mathfrak{JH}}$.
Clearly, $\zeta^{\prime}(\varphi)\zeta(\varphi)^{-1}=\varphi^{*}(x)x^{-1}$ so

$[\zeta]=[\zeta^{\prime}]\quad\mbox{in}\quad
H^{1}(\mathrm{Aut}(\mathsf{G}),\check{H}^{1}(\Gamma,U(1))).$

To prove the final remark, we note that Proposition 0.8.6 (and a computation
involving the definition of $\mathcal{H}_{\Gamma}$) implies that $\varphi$
lifts to $\mathcal{H}_{\Gamma}$ if and only if there exists a group cocycle
$(\tau_{\varphi,\boldsymbol{B}})_{\boldsymbol{B}\in\mathfrak{JH}}$ such that

$\varphi^{*}(u_{\boldsymbol{B},\boldsymbol{B}^{\prime}})u_{\varphi(\boldsymbol{B}),\varphi(\boldsymbol{B}^{\prime})}=\tau_{\varphi,\boldsymbol{B}}\tau_{\varphi,\boldsymbol{B}^{\prime}}^{-1}.$

In particular, $\varphi$ lifts to $\mathcal{H}_{\Gamma}$ if and only if
$\zeta(\varphi)=[\delta(\tau_{\varphi,\boldsymbol{B}})_{\boldsymbol{B}\in\mathfrak{JH}}]=0$.
The argument for the case when
$\zeta(\varphi)=\varphi^{*}(c_{1}(L))c_{1}(L^{-1})$, for a line bundle
$L\to\Gamma$, goes similarly. ∎

######  Proposition 0.8.8.

Let $\mathsf{G}$ be a simply connected nilpotent Lie group admitting flat
orbits. Assume that $\text{\mxedr.g}\subseteq\mathrm{Aut}(\mathsf{G})$ is a
subgroup. The following are equivalent

  1. (1)

$[\zeta|_{\text{\mxedr.g}}]=0\in
H^{1}(\text{\mxedr.g},\check{H}^{1}(\Gamma,U(1)))$

  2. (2)

Up to tensoring $\mathcal{H}_{\Gamma}$ with a line bundle $L\to\Gamma$
(solving $\zeta(\varphi)=\varphi^{*}(c_{1}(L))c_{1}(L^{-1})$ for each
$\varphi\in\text{\mxedr.g}$), each $\varphi\in\text{\mxedr.g}$ lifts to
$\mathcal{H}_{\Gamma}$ and the choice can locally in .g be made so that the
lift depends strongly continuously on $\varphi$.

###### Proof.

If $[\zeta|_{\text{\mxedr.g}}]=0\in
H^{1}(\text{\mxedr.g},\check{H}^{1}(\Gamma,U(1)))$, there exists an
$x\in\check{H}^{1}(\Gamma,U(1))$ with $\zeta(\varphi)=\varphi^{*}(x)x^{-1}$
for all $\varphi\in\text{\mxedr.g}$. There is a line bundle $L\to\Gamma$ with
$c_{1}(L)$ mapped to $x$ under
$\check{H}^{1}(\Gamma,U(1))\cong\check{H}^{2}(\Gamma,\mathbb{Z})$. By
Proposition 0.8.7, we can lift all $\varphi$ in a strongly continuous manner.
The converse follows in the same way. ∎

######  Definition 0.8.9.

Let $\mathsf{G}$ be a simply connected nilpotent Lie group admitting flat
orbits and $\text{\mxedr.g}\subseteq\mathrm{Aut}(\mathsf{G})$ a subgroup such
that $[\zeta|_{\text{\mxedr.g}}]=0\in
H^{1}(\text{\mxedr.g},\check{H}^{1}(\Gamma,U(1)))$.

  1. (1)

Let $L_{\text{\mxedr.g}}\to\Gamma$ be a line bundle such that
$\zeta(\varphi)=\varphi^{*}(c_{1}(L_{\text{\mxedr.g}}))c_{1}(L_{\text{\mxedr.g}})^{-1}$
for each $\varphi\in\text{\mxedr.g}$, and define

$\mathcal{H}_{\Gamma,L_{\text{\mxedr.g}}}:=\mathcal{H}_{\Gamma}\otimes
L_{\text{\mxedr.g}}.$

  2. (2)

For each $\varphi\in\text{\mxedr.g}$, pick lifts $\alpha(\varphi)$ to
$\mathcal{H}_{\Gamma,L_{\text{\mxedr.g}}}$ as in Proposition 0.8.8. Define the
map

$c_{\text{\mxedr.g}}:\text{\mxedr.g}\times\text{\mxedr.g}\to
C(\Gamma,U(1)),\quad(\varphi,\varphi^{\prime})\mapsto\alpha(\varphi)\varphi^{*}(\alpha(\varphi^{\prime}))\alpha(\varphi\varphi^{\prime})^{-1}.$

######  Proposition 0.8.10.

Let $\mathsf{G}$ be a simply connected nilpotent Lie group admitting flat
orbits. Assume that $\text{\mxedr.g}\subseteq\mathrm{Aut}(\mathsf{G})$ is a
subgroup such that $[\zeta|_{\text{\mxedr.g}}]=0\in
H^{1}(\text{\mxedr.g},\check{H}^{1}(\Gamma,U(1)))$ and construct
$\mathcal{H}_{\Gamma,L_{\text{\mxedr.g}}}$ and $c_{\text{\mxedr.g}}$ as in
Definition 0.8.9. Then $c_{\text{\mxedr.g}}$ satisfies the following:

  1. (1)

$c_{\text{\mxedr.g}}\in Z^{2}(\text{\mxedr.g},C(\Gamma,U(1)))$ is a group
cocycle for the induced .g-action on $C(\Gamma,U(1))$.

  2. (2)

The class $[c_{\text{\mxedr.g}}]\in H^{2}(\text{\mxedr.g},C(\Gamma,U(1)))$ is
independent of the choice of $L_{\text{\mxedr.g}}$ and lifts
$\alpha(\varphi)$.

  3. (3)

The lifts of .g to $\mathcal{H}_{\Gamma,L_{\text{\mxedr.g}}}$ can be choosen
to define a strongly continuous .g-action if and only if
$[c_{\text{\mxedr.g}}]=0\in H^{2}(\text{\mxedr.g},C(\Gamma,U(1)))$.

###### Proof.

Item 1 is a short computation. Item 2 follows from noting that the lift
$\alpha$ is unique up to a $C(\Gamma,U(1))$-factor, so two different choices
will give rise to cohomologous $c_{\text{\mxedr.g}}$.

To prove item 3, we first note that the lifts $\alpha(\varphi)$ are uniquely
determined by $\varphi$ up to a $C(\Gamma,U(1))$-factor. We have that
$[c_{\text{\mxedr.g}}]=0\in H^{2}(\text{\mxedr.g},C(\Gamma,U(1)))$ if and only
if there is a $\tau_{\text{\mxedr.g}}:\text{\mxedr.g}\to C(\Gamma,U(1))$ such
that

$c_{\text{\mxedr.g}}(\varphi,\varphi)=\tau_{\text{\mxedr.g}}(\varphi)\varphi^{*}(\tau_{\text{\mxedr.g}}(\varphi^{\prime}))\tau_{\text{\mxedr.g}}(\varphi\varphi^{\prime})^{-1}.$

Existence of such a $\tau_{\text{\mxedr.g}}$ is clearly equivalent to
existence of a modification of the lifts $\alpha(\varphi)$ by a
$C(\Gamma,U(1))$-factor such that
$\alpha(\varphi)\varphi^{*}\alpha(\varphi^{\prime})=\alpha(\varphi\varphi^{\prime})$.
∎

######  Theorem 0.8.11.

Let $\mathsf{G}$ be a simply connected nilpotent Lie group admitting flat
orbits and $\text{\mxedr.g}\subseteq\mathrm{Aut}(\mathsf{G})$. Then the
following are equivalent:

  1. (1)

There exists a Hilbert space bundle $\mathcal{H}\to\Gamma$ with a fibrewise
strongly continuous $\mathsf{G}$-action, with the unitary equivalence class of
the representation in the fibre over $\xi\in\Gamma$ is exactly $\xi$, such
that the .g-action on $\Gamma$ lifts to a strongly continuous action on
$\mathcal{H}\to\Gamma$ and a .g-equivariant $*$-isomorphism

$\pi_{\musFlat}:I_{\mathsf{G}}\to C_{0}(\Gamma,\mathbb{K}(\mathcal{H})).$

  2. (2)

It holds that $[\zeta|_{\text{\mxedr.g}}]=0\in
H^{1}(\text{\mxedr.g},\check{H}^{1}(\Gamma,U(1)))$ and
$[c_{\text{\mxedr.g}}]=0\in H^{2}(\text{\mxedr.g},C(\Gamma,U(1)))$.

  3. (3)

There exists a $C_{0}(\Gamma)$-linear .g-equivariant Morita equivalence

$I_{\mathsf{G}}\sim_{M}C_{0}(\Gamma).$

###### Proof.

That item 1 and 3 are equivalent follows from Kasparov’s stabilization theorem
and the definition of equivariant Morita equivalence. Clearly item 1 implies
item 2, and item 2 implies item 1 by Proposition 0.8.10 (with
$\mathcal{H}=\mathcal{H}_{\Gamma,L_{\text{\mxedr.g}}}$). ∎

######  Remark 0.8.12.

Let us take a moment to summarize the constructions above. To construct an
equivariant bundle of flat representations on a simply connected nilpotent Lie
group one first constructs $\mathcal{H}_{\Gamma}$ as in Proposition 0.8.5.
Secondly, if $\text{\mxedr.g}\subseteq\mathrm{Aut}(\mathsf{G})$ satisfies
$[\zeta|_{\text{\mxedr.g}}]=0\in
H^{1}(\text{\mxedr.g},\check{H}^{1}(\Gamma,U(1)))$ we can modify
$\mathcal{H}_{\Gamma}$ by a line bundle to
$\mathcal{H}_{\Gamma,L_{\text{\mxedr.g}}}$ as in Proposition 0.8.8 and lift
the action of .g to $\mathcal{H}_{\Gamma,L_{\text{\mxedr.g}}}$. If
$[c_{\text{\mxedr.g}}]=0\in H^{2}(\text{\mxedr.g},C(\Gamma,U(1))$, Proposition
0.8.10 allow us modify each $\alpha(\varphi)$, $\varphi\in\text{\mxedr.g}$, by
a scalar valued function such that .g acts on
$\mathcal{H}_{\Gamma,L_{\text{\mxedr.g}}}$.

######  Remark 0.8.13.

In [42], an equivariant Brauer group was defined. For a second countable group
.g acting on a second countable space $T$, the equivariant Brauer group
$\mathrm{Br}_{\text{\mxedr.g}}(T)$ is defined as the $T$-linear .g-equivariant
Morita equivalence classes of .g-continuous trace algebras $A$ over $T$. The
product in $\mathrm{Br}_{\text{\mxedr.g}}(T)$ is defined by
$\otimes_{C_{0}(T)}$ and the unit by $C_{0}(T)$. In [42, Theorem 5.1], it was
proven that there is a filtration $0\subseteq B_{1}\subseteq B_{2}\subseteq
B_{3}=\mathrm{Br}_{\text{\mxedr.g}}(T)$ and morphisms

$\displaystyle\delta_{\rm DD}:B_{3}=\mathsf{Br}_{\text{\mxedr.g}}(T)$
$\displaystyle\rightarrow\widecheck{H}^{3}(T,\mathbb{Z}),$
$\displaystyle\zeta:B_{2}=\ker\delta_{\rm DD}$ $\displaystyle\rightarrow
H^{1}(\text{\mxedr.g},\widecheck{H}^{1}(T,U(1))),$ $\displaystyle
c:B_{1}=\ker\zeta$ $\displaystyle\rightarrow
H^{2}(\text{\mxedr.g},C(T,U(1))),$

such that $A\sim_{M}C_{0}(T)$ (.g-equivariantly over $T$) if and only if
$\delta_{\rm DD}(A)=0\in H^{3}(T,\mathbb{Z})$, $\zeta(A)=0\in
H^{1}(\text{\mxedr.g},\check{H}^{1}(T,U(1)))$ and $c(A)=0\in
H^{2}(\text{\mxedr.g},C(T,U(1)))$. As such, Theorem 0.8.11 is a special case
of the results in [42].

The description above can be made substantially simpler in the special case
that there is a global bundle of polarizations over all of $\Gamma$. This
occurs for instance when the $\Gamma$ only consists of one fine strata or the
bundle of Vergne polarizations is a constant subbundle.

######  Proposition 0.8.14.

Let $\mathsf{G}$ be a simply connected nilpotent Lie group admitting flat
orbits. Assume that there exists a sub-bundle
$F\subseteq\Gamma\times\mathfrak{g}$ such that the fibre over $\xi\in\Gamma$
is a polarization of $\xi$. Then the following holds:

  * •

It holds that $[\zeta]=0\in
H^{1}(\mathrm{Aut}(\mathsf{G}),\check{H}^{1}(\Gamma,U(1)))$.

  * •

Define the associated Hilbert space bundle $\mathcal{H}_{F}\to\Gamma$, as in
Proposition 0.5.4, with a strongly continuous fibrewise $\mathsf{G}$-action
such that $[\mathcal{H}_{F,\xi}]=\xi$ for each $\xi\in\Gamma$. Then each
$\varphi\in\mathrm{Aut}(\mathsf{G})$ lifts to the morphism given by the
composition

$\mathcal{H}_{F}\xrightarrow{t_{F}(\varphi)}\mathcal{H}_{F_{\varphi}}\xrightarrow{\mathfrak{L}^{(0)}_{F,F_{\varphi}}}\mathcal{H}_{F}.$

where $F_{\varphi}=(\varphi^{*}\times\varphi^{-1})(F)\to\Gamma$ is the
transformed bundle of polarizations (cf. Proposition 0.8.6.a) and
$t_{F}(\varphi):\mathcal{H}_{F}\to\mathcal{H}_{F_{\varphi}}$ the map
$(\xi,f)\mapsto(\varphi^{*}\xi,f\circ\varphi^{*})$ (cf. Proposition 0.8.6.b)
and
$\mathfrak{L}^{(0)}_{F,F_{\varphi}}:\mathcal{H}_{F_{\varphi}}\to\mathcal{H}_{F}$
is the Lion transform (see Proposition 0.4.2).

  * •

The cocycle

$c_{\mathrm{Aut}(\mathsf{G})}:\mathrm{Aut}(\mathsf{G})\times\mathrm{Aut}(\mathsf{G})\to
C(\Gamma,U(1)),$

is the image of $c_{F,{\rm fl}}\in
Z^{2}(\mathrm{Aut}(\mathsf{G}),C(\Gamma,\mathbb{Z}/8))$ defined by

$c_{F,{\rm
fl}}(\varphi,\varphi^{\prime}):=\mathrm{Mas}(F_{\varphi},\varphi^{*}F_{\varphi^{\prime}},F_{\varphi\varphi^{\prime}})\\!\\!\\!\mod
8\mathbb{Z}.$

###### Proof.

The first item follows from the second item by Proposition 0.8.7. The second
and third item follows from Proposition 0.4.2. ∎

######  Proposition 0.8.15.

Let $\mathsf{G}$ be a simply connected nilpotent Lie group admitting flat
orbits. Let $d:=\mathrm{codim}(\mathfrak{z})/2$. Assume that there exists a
sub-bundle $F\subseteq\Gamma\times\mathfrak{g}$ such that the fibre over
$\xi\in\Gamma$ is a polarization of $\xi$ and
$T_{F}:\mathcal{H}_{F}\to\Gamma\times L^{2}(\mathbb{R}^{d})$ be a global
trivialization. Let $\text{\mxedr.g}\subseteq\mathrm{Aut}(\mathsf{G})$ be a
subgroup. Then the following are equivalent:

  1. (1)

$[c_{\rm
fl}|_{\text{\mxedr.g}}]\in\ker(H^{2}(\text{\mxedr.g},C(\Gamma,\mathbb{Z}/8))\to
H^{2}(\text{\mxedr.g},C(\Gamma,U(1))))$

  2. (2)

The natural homomorphism $\text{\mxedr.g}\to\mathrm{Aut}(I_{\mathsf{G}})\cong
C(\Gamma,PU(L^{2}(\mathbb{R}^{d})))$ (via $\mathrm{Ad}(T_{F})$) lifts to a
homomorphism $\text{\mxedr.g}\to C(\Gamma,U(L^{2}(\mathbb{R}^{d})))$.

  3. (3)

The .g-action on $I_{\mathsf{G}}$ is induced from a strongly continuous
.g-action on $\mathcal{H}_{F}$.

###### Proof.

The proposition follows from Proposition 0.8.10 and 0.8.14. ∎

###  0.9. Examples of graded nilpotent Lie groups

In this subsection we give some salient examples of simply connected,
nilpotent Lie groups geared towards the geometric scenarios arising later in
the monograph. The examples aim at exemplifying the contents of the recent
subsections with a focus on describing gradings, computing the top strata
$\Gamma$, the Hilbert space bundle $\mathcal{H}_{\Gamma}\to\Gamma$, the ideal
$I_{\mathsf{G}}\subseteq C^{*}(\mathsf{G})$ and in particular how to single
out a compact subgroup $\text{\mxedr.g}\subseteq\mathrm{Aut}(\mathsf{G})$
lifting to an action on $\mathcal{H}_{\Gamma}$.

####  0.9.1. Step 1 graded nilpotent Lie groups

Nilpotent Lie groups of step length 1 are just the abelian Lie groups. So the
term simply connected, nilpotent, graded Lie group of step length 1 is a
lengthy description of the Lie group $\mathsf{G}=\mathbb{R}^{n}$, for some
$n\in\mathbb{N}$, equipped with a vector space grading

$\mathfrak{g}\equiv\mathbb{R}^{n}=\bigoplus_{p<0}\mathbb{R}^{n_{p}}[p],$

where we write $\mathbb{R}^{n_{p}}[p]$ to indicate the space
$\mathbb{R}^{n_{p}}$ where the grading on this subspace is $p$. Clearly any
such decomposition with $\sum_{p<0}n_{p}=n$ defines a grading of the Lie
algebra $\mathbb{R}^{n}$. Since $\mathsf{G}=\mathbb{R}^{n}$ is abelian, we
have that

$\mathrm{Aut}(\mathsf{G})=GL_{n}(\mathbb{R})\quad\mbox{and}\quad\mathrm{Aut}_{\rm
gr}(\mathsf{G})=\bigoplus_{p<0}GL_{n_{p}}(\mathbb{R})$

We can also conclude that

$\hat{\mathsf{G}}=\mathfrak{g}^{*}=\Gamma=\Xi,$

which only consists of one strata. In particular,
$C^{*}(\mathsf{G})=I=C_{0}(\mathfrak{g}^{*})=C_{0}(\Xi)=C_{0}(\Gamma)$. We
have that $\mathcal{H}_{V}=\Gamma\times\mathbb{C}$ is a trivial line bundle
and the $\mathrm{Aut}(\mathsf{G})$-action on $I$ lifts to an action on
$\mathcal{H}_{V}$ which is trivial on the fibre, so in particular $\delta_{\rm
DD}(I_{\mathsf{G}})=0$, $\zeta=0$, and $[c_{\rm flat}]=0\in
H^{2}(GL_{n}(\mathbb{R}),C_{0}(\Gamma,\mathbb{C}^{\times}))$. This example is
likely too trivial to improve the understanding of the reader, as the whole
theory collapses to the standard Fourier transform on a graded vector space,
but we have included it as it contains the local model for foliations as a
special case (see more below in Example 0.16.2) and isolates the effects of
choosing a grading.

####  0.9.2. Step 2 graded nilpotent Lie groups

Let us consider the case of step length $2$. This situation arises
geometrically when for instance having a non-integrable length 2 filtration of
a manifold, i.e. one specified subbundle of the tangent bundle which is not
integrable.

If $\mathfrak{g}$ is a nilpotent Lie algebra of step length $2$, then
$\overline{\mathfrak{g}}:=\mathfrak{g}/[\mathfrak{g},\mathfrak{g}]$ is an
abelian Lie algebra. In particular, $\mathfrak{g}$ as a Lie algebra is
determined by a subspace $C(\mathfrak{g})\subseteq\mathfrak{g}$ and a
surjective map $\omega:\overline{\mathfrak{g}}\wedge\overline{\mathfrak{g}}\to
C(\mathfrak{g})$, where
$\overline{\mathfrak{g}}:=\mathfrak{g}/C(\mathfrak{g})$, by

$[X,Y]=\omega(\overline{X},\overline{Y}),$

where we use the notation $\overline{X}:=X+C(\mathfrak{g})$. Surjectivity of
$\omega$ ensures that $[\mathfrak{g},\mathfrak{g}]=C(\mathfrak{g})$. We say
that $\mathfrak{g}$ is a 2-step nilpotent lie algebra of type $(p,q)$ if

$\dim[\mathfrak{g},\mathfrak{g}]=p\quad\mbox{and}\quad\dim\overline{\mathfrak{g}}=q.$

For more details, see [62]. Following Remark 0.6.7, we let
$X_{1},\ldots,X_{p}$ denote a basis of $[\mathfrak{g},\mathfrak{g}]$ and we
define
$\overline{\omega}_{k}:\overline{\mathfrak{g}}\wedge\overline{\mathfrak{g}}\to\mathbb{R}$,
$k=1,\ldots,p$, by

$[X,Y]=\sum_{k=1}^{p}\overline{\omega}_{k}(\overline{X},\overline{Y})X_{k}.$

Surjectivity of $\omega$ ensures that the forms
$\overline{\omega}_{1},\ldots,\overline{\omega}_{p}$ are linearly independent
in $(\overline{\mathfrak{g}}\wedge\overline{\mathfrak{g}})^{*}$. The basis
induces an inner product on $\mathfrak{g}$ and we can identify
$\overline{\mathfrak{g}}$ with the orthogonal complement of
$C(\mathfrak{g})\subseteq\mathfrak{g}$ and
$(\overline{\mathfrak{g}}\wedge\overline{\mathfrak{g}})^{*}$ with
$\mathfrak{so}(\overline{\mathfrak{g}})$. We identify the collection
$\overline{\omega}_{1},\ldots,\overline{\omega}_{p}\in(\overline{\mathfrak{g}}\wedge\overline{\mathfrak{g}})^{*}$
with a collection
$\omega^{1},\ldots,\omega^{p}\in\mathfrak{so}(\overline{\mathfrak{g}})$ and
let $W\subseteq\mathfrak{so}(\overline{\mathfrak{g}})$ denote the space
spanned by $\omega^{1},\ldots,\omega^{p}$.

For an arbitrary finite-dimensional inner product space $V$ and a subspace
$W\subseteq\mathfrak{so}(V)$, we can equip $V\oplus W$ with a Lie bracket by
declaring $W$ to be central and for $X,Y\in V$ the element $[X,Y]\in W$ is
given by that

$([X,Y],Z)_{\mathfrak{so}(V)}=(Z(X),Y)_{V}\quad\forall Z\in W.$

We denote this Lie algebra by $V_{W}$. It is readily verified that $V_{W}$ is
a nilpotent Lie algebra of step length $2$ of type $(\dim(W),\dim(V))$ with
$C(V_{W})=W$. The reader should note that $V_{\mathfrak{so}(V)}$ is the free
step 2 nilpotent Lie algebra on $V$ and that for any
$W\subseteq\mathfrak{so}(V)$ the following sequence is exact

$0\to W^{\perp}\to V_{\mathfrak{so}(V)}\to V_{W}\to 0.$ (16)

It was proven in [62] that this construction up to isomorphism exhausts all
nilpotent Lie algebras of step length $2$.

######  Proposition 0.9.1 (Proposition 3.1.1 of [62]).

Let $\mathfrak{g}$ be a step $2$ nilpotent Lie algebra and type $(p,q)$ with a
choice of inner product $(\cdot,\cdot)_{\overline{\mathfrak{g}}}$ on
$\overline{\mathfrak{g}}$ and a Jordan-Hölder basis $X_{1},\ldots,X_{n}$ with
$\mathfrak{g}_{p}=C(\mathfrak{g})$. Define
$\omega^{1},\ldots,\omega^{p}\in\mathfrak{so}(\overline{\mathfrak{g}})$ and
$W\subseteq\mathfrak{so}(\overline{\mathfrak{g}})$ as above. For the basis
$w_{1},\ldots,w_{p}\in W$ such that
$(\omega^{i},w_{j})_{\mathfrak{so}(\overline{\mathfrak{g}})}=\delta_{i,j}$,
the mapping

$\mathfrak{g}\to\overline{\mathfrak{g}}_{W},\quad
X_{j}\mapsto\begin{cases}w_{j},\;j=1,\ldots,p,\\\
\overline{X}_{j},\;j=p+1,\ldots,n,\end{cases}$

is a Lie algebra isomorphism.

######  Remark 0.9.2.

We note that Proposition 0.9.1 implies that for a step $2$ nilpotent Lie
algebra $\mathfrak{g}$, it holds that

$\mathfrak{z}=\\{X\in\mathfrak{g}:w(\overline{X})=0\forall w\in W\\},$

where $\mathfrak{z}$ denotes the center of $\mathfrak{g}$. In particular,
$C(\mathfrak{g})=\mathfrak{z}$ if and only if for any
$X\in\overline{\mathfrak{g}}$, there is a $w\in W$ with $w(X)\neq 0$.

From this characterization of step $2$ nilpotent Lie algebras, [62] ensures an
explicit description of $\mathrm{Aut}(\mathfrak{g})$. For a step $2$ nilpotent
Lie algebra $\mathfrak{g}$ with a choice of inner product on
$\overline{\mathfrak{g}}$, with
$W\subseteq\mathfrak{so}(\overline{\mathfrak{g}})$ as above, we define the Lie
group

$H_{\mathfrak{g}}:=\\{g\in GL(\overline{\mathfrak{g}}):g^{t}Wg\subseteq W\\}.$

By universality, the group $GL(\overline{\mathfrak{g}})$ acts on
$\overline{\mathfrak{g}}_{\mathfrak{so}(\overline{\mathfrak{g}})}$ – the free
step 2 nilpotent Lie algebra on $\overline{\mathfrak{g}}$. We define
$u_{H}:H_{\mathfrak{g}}\to\mathrm{Aut}(\mathfrak{g})$ by noting that the short
exact sequence (16) ensures that the $H_{\mathfrak{g}}$-action descends to
$\mathfrak{g}\cong\overline{\mathfrak{g}}_{\mathfrak{so}(\overline{\mathfrak{g}})}/W^{\perp}$.
The vector space $\mathrm{Hom}(\overline{\mathfrak{g}},C(\mathfrak{g}))$ acts
as Lie algebra automorphisms on $\mathfrak{g}$ by $u_{t}(A)(v\oplus
w):=v\oplus(Av+w)$ for $v\in\overline{\mathfrak{g}}$, $w\in C(\mathfrak{g})$
and $A\in\mathrm{Hom}(\overline{\mathfrak{g}},C(\mathfrak{g}))$. It is readily
verified that for $g\in H_{\mathfrak{g}}$ and
$A\in\mathrm{Hom}(\overline{\mathfrak{g}},C(\mathfrak{g}))$,

$u_{H}(g)u_{t}(A)u_{H}(g)^{-1}=u_{t}(g.A),$

where

$(g.A)(v):=gA(gv)g^{t}.$ (17)

Therefore $u_{t}$ and $u_{H}$ induces a well defined group homomorphism
$u_{t}\rtimes
u_{H}:\mathrm{Hom}(\overline{\mathfrak{g}},C(\mathfrak{g}))\rtimes
H_{\mathfrak{g}}\to\mathrm{Aut}(\mathfrak{g})$ where the crossed product is
with respect to the $H_{g}$-action on
$\mathrm{Hom}(\overline{\mathfrak{g}},C(\mathfrak{g}))$ given in (17).

######  Proposition 0.9.3 (Proposition 3.4.2 of [62]).

Let $\mathfrak{g}$ be a step $2$ nilpotent Lie algebra with a choice of inner
product on $\overline{\mathfrak{g}}:=\mathfrak{g}/C(\mathfrak{g})$. Then

$u_{t}\rtimes
u_{H}:\mathrm{Hom}(\overline{\mathfrak{g}},C(\mathfrak{g}))\rtimes
H_{\mathfrak{g}}\to\mathrm{Aut}(\mathfrak{g}),$

is a group isomorphism.

######  Remark 0.9.4.

The moduli space of step 2 nilpotent Lie algebras of type $(p,q)$ can by [61]
be identified with $G(p,\mathfrak{so}(q,\mathbb{R}))/GL_{q}(\mathbb{R})$,
where $G(p,\mathfrak{so}(q,\mathbb{R}))$ denotes the Grassmannian manifold of
$p$-dimensional subspaces in $\mathfrak{so}(q,\mathbb{R})$. The automorphism
group of a step 2 nilpotent Lie algebra of type $(p,q)$ is quite small (see
[62, Proposition 3.4.3]) for most values of $p$ and $q$ in the generic case
(for a dense open subset of the moduli space). In particular, for most values
of $p$ and $q$, step 2 nilpotent Lie algebras of type $(p,q)$ generically do
not admit gradings.

The geometric setting our Lie algebras arise from ensures that the Lie
algebras are graded which changes what should be meant with generic. Let us
describe possible gradings on step 2 nilpotent Lie algebras.

######  Proposition 0.9.5.

Let $\mathfrak{g}$ be a step $2$ nilpotent Lie algebra. If $\delta$ is a
dilation on $\mathfrak{g}$ it is uniquely determined by the induced dilation
$\overline{\delta}$ on $\overline{\mathfrak{g}}$ from the isomorphism
$\mathfrak{g}\cong\overline{\mathfrak{g}}_{W}=\overline{\mathfrak{g}}\oplus
C(\mathfrak{g})$ (from Proposition 0.9.1) and the identity

$\delta_{t}[X,Y]=\omega(\overline{\delta}_{t}(\overline{X}),\overline{\delta}_{t}(\overline{Y})),\quad\mbox{for
$[X,Y]\in C(\mathfrak{g})$}.$

For any grading on $\mathfrak{g}$, we have that $\mathrm{Aut}_{\rm
gr}(\mathfrak{g})\subseteq u_{H}(H_{\mathfrak{g}})$ (see Proposition 0.9.3 for
notation).

Moreover, a dilation $\overline{\delta}$ on $\overline{\mathfrak{g}}$ is
induced from a dilation $\delta$ on $\mathfrak{g}$ if and only if the grading
$\overline{\mathfrak{g}}=\bigoplus_{p<0}\overline{\mathfrak{g}}_{p}$
constructed from $\overline{\delta}$ (as in Proposition 0.7.3) satisfies that

$\left(\sum_{k+l<p}\omega(\overline{\mathfrak{g}}_{k},\overline{\mathfrak{g}}_{l})\right)\bigcap\left(\sum_{k+l=p}\omega(\overline{\mathfrak{g}}_{k},\overline{\mathfrak{g}}_{l})\right)=0\quad\forall
p<0.$ (18)

###### Proof.

The first part of the proposition follows from that a dilation is a
homomorphism $\delta:\mathbb{R}_{+}\to\mathrm{Aut}(\mathfrak{g})$ and the Lie
bracket defines a surjection
$\omega:\overline{\mathfrak{g}}\wedge\overline{\mathfrak{g}}\to
C(\mathfrak{g})$.

The second part follows from noting that the condition (18) is equivalent to
that the spaces

$C(\mathfrak{g})_{p}:=\sum_{k+l=p}\omega(\overline{\mathfrak{g}}_{k},\overline{\mathfrak{g}}_{l}),$

define a decomposition

$C(\mathfrak{g})=\bigoplus_{p<0}C(\mathfrak{g})_{p}.$

We can therefore define a dilation as in Proposition 0.7.3 from grading
$\mathfrak{g}$ by

$\mathfrak{g}_{p}:=\overline{\mathfrak{g}}_{p}\oplus C(\mathfrak{g})_{p}.$

The decomposition $\mathfrak{g}=\bigoplus_{p<0}\mathfrak{g}_{p}$ is indeed a
grading as it follows from the construction that

$[\mathfrak{g}_{p},\mathfrak{g}_{q}]=\omega(\overline{\mathfrak{g}}_{p},\overline{\mathfrak{g}}_{q})\subseteq
C(\mathfrak{g})_{p+q}\subseteq\mathfrak{g}_{p+q}.$

∎

We now turn our attention to representation theory of a step $2$ simply
connected nilpotent Lie group. We denote the Lie group by $\mathsf{G}$ and its
Lie algebra by $\mathfrak{g}$. As above we fix a Jordan-Hölder basis with
$\mathfrak{g}_{p}=C(\mathfrak{g})$ and
$\mathfrak{g}_{\dim(\mathfrak{z})}=\mathfrak{z}$. We let
$\omega_{1},\ldots,\omega_{p}:(\mathfrak{g}/\mathfrak{z})\wedge(\mathfrak{g}/\mathfrak{z})\to\mathbb{R}$
denote the forms induced from
$\overline{\omega}_{1},\ldots,\overline{\omega}_{p}:\overline{\mathfrak{g}}\wedge\overline{\mathfrak{g}}\to\mathbb{R}$.
The Pfaffian function (see Proposition 0.6.6 and Remark 0.6.7) on
$\mathfrak{z}^{*}$ can be described in the coordinates of the Jordan-Hölder
basis as the polynomial of degree $\mathrm{codim}(\mathfrak{z})/2$ given by

$\mathsf{Pf}(\xi)=\mathsf{Pf}\left(\sum_{j=1}^{p}\xi(X_{j})\omega_{j}\right).$

As such, if $\mathsf{G}$ admits flat orbits (i.e. the Pfaffian polynomial is
non-vanishing on $\mathfrak{z}^{*}$), the top stratum
$\Gamma\subseteq\hat{\mathsf{G}}$ can be described by the transversal
$\Lambda\subseteq\mathfrak{z}^{*}$ which is given as follows:

$\displaystyle\Lambda$
$\displaystyle=\left\\{\xi=\sum_{j=1}^{\dim(\mathfrak{z})}\xi(X_{j})\xi_{j}\in\mathfrak{z}^{*}:\mathsf{Pf}\left(\sum_{j=1}^{p}\xi(X_{j})\omega_{j}\right)\neq
0\right\\}=$ (19)
$\displaystyle=\left\\{\xi=\sum_{j=1}^{\dim(\mathfrak{z})}\xi(X_{j})\xi_{j}\in\mathfrak{z}^{*}:\det\left(\sum_{j=1}^{p}\xi(X_{j})\omega_{j}\right)\neq
0\right\\}=\Lambda_{C}\times(\mathfrak{z}/C(\mathfrak{g}))^{*},$

where

$\Lambda_{C}=\left\\{\xi=\sum_{j=1}^{p}\xi(X_{j})\xi_{j}\in
C(\mathfrak{g})^{*}:\mathsf{Pf}\left(\sum_{j=1}^{p}\xi(X_{j})\omega_{j}\right)\neq
0\right\\}.$

######  Remark 0.9.6.

If $C(\mathfrak{g})=\mathfrak{z}$, i.e.
$\overline{\mathfrak{g}}=\mathfrak{g}/\mathfrak{z}$, then we can also write

$\Lambda=w_{\mathfrak{z}}^{-1}(W\cap GL(\overline{\mathfrak{g}})),$ (20)

where $w_{\mathfrak{z}}:\mathfrak{z}^{*}\to W$ is defined by
$w_{\mathfrak{z}}(\xi):=\sum_{j=1}^{\dim(\mathfrak{z})}\xi(X_{j})\omega^{j}$.
We note that by Remark 0.9.2, $C(\mathfrak{g})=\mathfrak{z}$ if and only if
for any $X\in g$, $w(\overline{X})=0$ for all $w\in W$ implies that $X\in
C(\mathfrak{g})$. In light of Equation (20), and injectivity of the $W$-action
on $\mathfrak{g}$, it is tempting to suspect that the equality
$C(\mathfrak{g})=\mathfrak{z}$ implies the existence of flat orbits, but fails
since even if $C(\mathfrak{g})=\mathfrak{z}$ there need not exist an
individual element of $W\subseteq\mathfrak{so}(\overline{\mathfrak{g}})$
acting injectively on $\overline{\mathfrak{g}}$. This fails in the example of
Remark 0.9.19 below.

######  Theorem 0.9.7.

Let $\mathsf{G}$ be a step 2 simply connected nilpotent Lie group of type
$(p,q)$ admitting flat orbits. Set $d:=\mathrm{codim}(\mathfrak{z})/2$. We
consider
$\text{\mxedr.g}=u_{H}(H_{\mathfrak{g}})\subseteq\mathrm{Aut}(\mathsf{G})$.
Then the following holds for the ideal of flat orbits:

  1. (1)

It holds that $[\zeta|_{\text{\mxedr.g}}]=0\in
H^{1}(\text{\mxedr.g},\check{H}^{1}(\Gamma,U(1)))$.

  2. (2)

It holds that $[c_{\text{\mxedr.g}}]$ is the image of the analogous cohomology
class $[c_{GL(q)}]$, defined for the free step 2 simply connected nilpotent
Lie group on a $q$-dimensional vector space, under the mapping

$H^{2}(GL(q),C(GL(q)/Sp(q),U(1)))\to H^{2}(\text{\mxedr.g},C(\Gamma,U(1))),$

defined from the inclusion $\text{\mxedr.g}\hookrightarrow GL(q)$ and the
inclusion $\Gamma\hookrightarrow GL(q)\cap\mathfrak{so}(q)\cong GL(q)/Sp(q)$.

###### Proof.

Item 1) follows from that $[\zeta|_{\text{\mxedr.g}}]$ is the pullback of

$[\zeta|_{GL(q)}]\in H^{1}(GL(q),\check{H}^{1}(GL(q)/Sp(q),U(1)))=0,$

defined from the quotient map onto $\mathfrak{g}$ from the free step 2
nilpotent Lie algebra on a $q$-dimensional vector space. Here
$GL(q)\cap\mathfrak{so}(q)\cong GL(q)/Sp(q)$ is the space of flat orbits of
the free step 2 nilpotent Lie algebra on a $q$-dimensional vector space.
Indeed, by homotopy invariance we have that
$\check{H}^{1}(GL(q)/Sp(q),U(1))\cong\mathbb{Z}$ and
$H^{1}(GL(q),\mathbb{Z})=0$. Item 2) is now clear from construction. ∎

We let $\overline{G}=G/[G,G]$ denote abelianization of $G$ which coincides
with the simply connected Lie group associated with $\overline{\mathfrak{g}}$.
We identify $\overline{G}$ with $\overline{\mathfrak{g}}$. Pick a Jordan-
Hölder basis $\boldsymbol{B}\in\mathfrak{JH}$. By the proof of Proposition
0.5.2, $(X_{j})_{j\in K_{\boldsymbol{B}}}$ spans a complement to the Vergne
polarization $\mathfrak{h}_{V,\boldsymbol{B}}(\xi)$ for any $\xi\in
U_{\boldsymbol{B}}\subseteq\Lambda$. Write
$K_{\boldsymbol{B}}=\\{j_{1}<\cdots<j_{d}\\}$ where
$d=\mathrm{codim}(\mathfrak{z})/2$. From this fact we deduce the following
proposition.

######  Proposition 0.9.8.

Let $G$ be a step $2$ nilpotent Lie group admitting flat orbits. Fix a Jordan-
Hölder basis $\boldsymbol{B}\in\mathfrak{JH}$ with
$K_{\boldsymbol{B}}=\\{j_{1}<\cdots<j_{d}\\}$, where
$d=\mathrm{codim}(\mathfrak{z})/2$. Define the mapping
$\rho_{\boldsymbol{B}}:\mathbb{R}^{d}\to G$ by

$\rho_{\boldsymbol{B}}(x_{1},\ldots,x_{d}):=\exp\left(\sum_{i=1}^{d}x_{i}X_{j_{i}}\right).$

Then it holds that

  * •

The composition
$\mathbb{R}^{d}\xrightarrow{\rho_{\boldsymbol{B}}}G\to\overline{G}$ is linear.

  * •

For each $\xi\in U_{\boldsymbol{B}}\subseteq\Lambda$, the composition
$\mathbb{R}^{d}\xrightarrow{\rho_{\boldsymbol{B}}}G\to
G/H_{V,\boldsymbol{B}}(\xi)$ is a polynomial diffeomorphism.

######  Proposition 0.9.9.

Let $G$ be a step $2$ nilpotent Lie group admitting flat orbits with a Jordan-
Hölder basis $\boldsymbol{B}\in\mathfrak{JH}$. For $\xi\in
U_{\boldsymbol{B}}\subseteq\Lambda$, the mapping

$T_{V,\boldsymbol{B}}(\xi):\mathcal{H}_{F_{V,\boldsymbol{B}}(\mathcal{O},\xi),\xi}(\mathcal{O}_{\xi})\to
L^{2}(\mathbb{R}^{d}),\quad
T_{V,\boldsymbol{B}}(\xi)f(x):=f(\rho_{\boldsymbol{B}}(x)\xi),$

is a well defined unitary that fits in a unitary trivialization of Hilbert
space bundles over $\Gamma$

$T_{V,\boldsymbol{B}}:\mathcal{H}_{V,\boldsymbol{B}}\to
U_{\boldsymbol{B}}\times L^{2}(\mathbb{R}^{d}),\quad
T_{V,\boldsymbol{B}}(\xi,f):=(\xi,T_{V,\boldsymbol{B}}(\xi)f).$

The $G$-action on the trivial Hilbert space bundle $U_{\boldsymbol{B}}\times
L^{2}(\mathbb{R}^{d})$ induced by $T_{V,\boldsymbol{B}}$ and the $G$-action on
$\mathcal{H}_{V,\boldsymbol{B}}$ (see Proposition 0.5.4) takes the form of a
continuous homomorphism $\tilde{\pi}_{\boldsymbol{B}}:G\to
C(U_{\boldsymbol{B}},U(L^{2}(\mathbb{R}^{d})))$ where

$\tilde{\pi}_{\boldsymbol{B},\xi}(g)f(x)=\mathrm{e}^{i\xi(\log^{\prime}_{\xi}(g)+[\log(g),\log(\rho(x))])}f(x+\log_{\xi}^{0}(g)),\quad\forall
g\in G,\;\xi\in U_{\boldsymbol{B}},\;f\in L^{2}(\mathbb{R}^{d}),$

where we write $\log^{0}_{\xi}(g)$ for the projection of $\log(g)$ onto the
linear span of $(X_{j})_{j\in K_{\boldsymbol{B}}}$ (along the Vergne
polarization) and
$\log^{\prime}_{\xi}(g):=\log(g)-\log^{0}_{\xi}(g)\in\mathfrak{h}_{V,\boldsymbol{B}}(\xi)$.

This proposition is a special case of Lemma 0.5.5.

###### Proof.

The statements concerning $T_{V}$ follows from inspecting the proof of
Proposition 0.5.4. Let us turn to the form of the representation
$\tilde{\pi}$. Note that $\log\rho(x)=\sum_{i=1}^{d}x_{i}X_{j_{i}}$. For
$X=X_{0}+X^{\prime}\in\mathfrak{g}$, with $X_{0}$ being the projection of $X$
onto the linear span of $(X_{j})_{j\in K_{\boldsymbol{B}}}$ (along the Vergne
projection) and $X^{\prime}\in\mathfrak{h}_{V,\boldsymbol{B}}(\xi)$, and
$x\in\mathbb{R}^{d}$, we have that

$\displaystyle\mathrm{e}^{X}\rho(x)=\mathrm{e}^{X_{0}+X^{\prime}}\mathrm{e}^{\log\rho(x)}$
$\displaystyle=\mathrm{e}^{X_{0}+X^{\prime}+\log\rho(x)+\frac{1}{2}[X,\log\rho(x)]}=$
$\displaystyle=\mathrm{e}^{X_{0}+\log\rho(x)}\mathrm{e}^{X^{\prime}+[X,\log\rho(x)]}=$
$\displaystyle=\rho(x+X_{0})\mathrm{e}^{X^{\prime}+[X,\log\rho(x)]},$

where we have identified $X_{0}$ with its coordinates in $\mathbb{R}^{d}$ with
respect to $(X_{j})_{j\in K_{\boldsymbol{B}}}$. Since $\mathfrak{g}$ is of
step length $2$,
$[X,\log\rho(x)]\in\mathfrak{z}\subseteq\mathfrak{h}_{V,\boldsymbol{B}}(\xi)$
so $\mathrm{e}^{X^{\prime}+[X,\log\rho(x)]}\in H_{V,\boldsymbol{B}}(\xi)$.
Therefore the representation $\tilde{\pi}_{\boldsymbol{B}}$ takes the stated
form. ∎

######  Remark 0.9.10.

The explicit description of the representations of flat orbits for step length
$2$ from Proposition 0.9.9 allow us to define a “noncommutative Fourier
transform” on flat orbits $\mathsf{F}_{\boldsymbol{B}}:C^{*}(G)\to
C_{b}(U_{\boldsymbol{B}},\mathbb{K}(L^{2}(\mathbb{R}^{d})))$ as the right
vertical morphism of Proposition 12. By Remark 0.5.10, this “noncommutative
Fourier transform” on flat orbits is injective. It can be defined as the
closure of the linear map $\mathsf{F}_{\boldsymbol{B}}:\mathcal{S}(G)\to
C_{b}(U_{\boldsymbol{B}},\mathcal{S}(\mathbb{R}^{d}\times\mathbb{R}^{d}))$
given by

$[\mathsf{F}_{\boldsymbol{B}}a](\xi,x,y):=\int_{\mathfrak{h}_{V,\boldsymbol{B}}(\xi)}[\mathrm{exp}^{*}a](y-x,X^{\prime})\mathrm{e}^{i\xi(X^{\prime}+[X^{\prime}-y,x])}\mathrm{d}X^{\prime}$

where we identify $\mathbb{R}^{d}$ with the linear span of $(X_{j})_{j\in
K_{\boldsymbol{B}}}$. We note that the “noncommutative Fourier transform” on
flat orbits and Equation 19 produces the isomorphism

$I_{G,\boldsymbol{B}}\cong C_{0}((\mathfrak{z}/C(\mathfrak{g}))^{*}\cap
U_{\boldsymbol{B}})\otimes C_{0}(\Lambda_{C}\cap
U_{\boldsymbol{B}},\mathbb{K}(L^{2}(\mathbb{R}^{d}))),$

and on the first tensor factor the noncommutative Fourier transform acts just
as the ordinary Euclidean Fourier transform.

We now turn our attention to more explicit examples.

######  Example 0.9.11.

The most fundamental example of a step 2 nilpotent Lie algebra is the
Heisenberg group. The Heisenberg group is the local model for contact
manifolds (see more in Example 0.16.5 below). The reader can also consult [71]
for a broader view on the Heisenberg group. If $V$ is a finite dimensional
real vector space with a non-degenerate 2-form $\omega$ and basis vector
$Z\in\mathbb{R}$, there is a Lie bracket on $V\oplus\mathbb{R}$ such that its
centre is the copy of $\mathbb{R}$, and for $X,Y\in V$

$[X,Y]:=\omega(X,Y)Z.$

We often suppress the role of $Z$, and simply denote the associated Lie
algebra by $V\ltimes_{\omega}\mathbb{R}$. It is clear from the construction
that $V\ltimes_{\omega}\mathbb{R}$ is step $2$ nilpotent of type
$(1,\dim(V))$. A choice of inner product on $V$ induces an isomorphism
$V\ltimes_{\omega}\mathbb{R}\cong V_{\mathbb{R}\omega^{1}}$ by Proposition
0.9.1. There is a standard grading on $V\ltimes_{\omega}\mathbb{R}$ given by

$(V\ltimes_{\omega}\mathbb{R})_{-1}:=V\quad\mbox{and}\quad(V\ltimes_{\omega}\mathbb{R})_{-2}:=\mathbb{R}.$

We use the notation $\mathfrak{h}_{n}$ for a Heisenberg Lie algebra on a
$2n$-dimensional vector space equipped with a non-degenerate $2$-form and
$\mathbb{H}_{n}$ for the associated step $2$ simply connected nilpotent Lie
group. Up to (non-canonical) isomorphism, $\mathfrak{h}_{n}$ and
$\mathbb{H}_{n}$ only depend on $n$.

We conclude from Proposition 0.9.3 that

$\mathrm{Aut}(\mathfrak{h}_{n})=V^{*}\rtimes((Sp(V,\omega)\rtimes\mathbb{Z}/2)\times\mathbb{R}_{>0}).$

See also [71, Chapter I.2]. Here we can identify
$Sp(V,\omega)\rtimes\mathbb{Z}/2$ with the elements $g\in GL(V)$ such that
$g\omega g^{t}=\pm\omega$ and the generator of $\mathbb{Z}/2$ is an order two
element $g$ such that $g\omega g^{t}=-\omega$ – it can be taken as $i$ times
complex conjugation upon choosing an isomorphism of $V$ with
$\mathbb{C}^{\dim(V)/2}$ transforming $\omega$ to the standard symplectic form
on $\mathbb{C}^{\dim(V)/2}=\mathbb{R}^{\dim(V)}$. The graded automorphisms,
for the standard grading, are described similarly as

$\mathrm{Aut}_{\rm
gr}(\mathfrak{h}_{n})=(Sp(V,\omega)\rtimes\mathbb{Z}/2)\times\mathbb{R}_{>0}.$

For the sake of simplicity, we fix a complex structure on $V$ adapted to
$\omega$. This induces an inner product structure on $V$, and we can (and
will) identify $\omega$ with the associated element in $\mathfrak{so}(V)$.
Since $\omega$ is non-degenerate, Equation (19) implies that

$\Gamma=\\{\xi\in\mathfrak{z}^{*}:\xi\neq 0\\},$

and $\Gamma$ consists of one fine strata. In fact, the equality
$\Gamma=\mathfrak{z}^{*}\setminus\\{0\\}$ holds for all Lie groups admitting
flat orbits and whose center is one-dimensional (see below in Subsection
0.9.3). We note that the action

$\mathrm{Aut}(\mathfrak{h}_{n})\times\Gamma\to\Gamma,$

of the automorphisms on the flat orbits (see Proposition 0.7.11) factors over
the quotient to the unique dilation on the Heisenberg Lie algebra

$\mathrm{Aut}(\mathfrak{h}_{n})=V^{*}\rtimes((Sp(V,\omega)\rtimes\mathbb{Z}/2)\times\mathbb{R}_{>0})\to\mathbb{R}_{>0}.$

By Proposition 0.7.6, we can identify

$\Gamma_{\partial}=\\{\xi\in\mathfrak{z}^{*}:\xi(Z)=\pm 1\\}.$

For the Heisenberg group, $\widehat{\mathbb{H}_{n}}=\Gamma\cup\Gamma_{2}$ (as
sets) where $\Gamma_{2}=V^{*}$ are the characters. Therefore, the
representation theory is completely described once having described the
representations of flat orbits. We fix a complex structure $J$ on $V$ adapted
to $\omega$ and consider $V$ as an inner product space. Set $d:=\dim(V)/2$.
Note that a subspace $V_{0}\subseteq V$ is Lagrangian (i.e. $V_{0}=\\{X\in
V:\omega(X,Y)=0\;\forall Y\in V_{0}\\}$) if and only if $V_{0}$ and $JV_{0}$
are orthogonal and $V=V_{0}+JV_{0}$. Fix such a subspace $V_{0}$, then for any
basis $X_{1},\ldots,X_{d}$ we obtain a Jordan-Hölder basis
$Z,X_{1},\ldots,X_{1},Y_{1},\ldots,Y_{d}$ where $Y_{j}:=JX_{j}$. If
$X_{1},\ldots,X_{d}$ is an ON-basis, we have the well known Heisenberg
relations

$[X_{j},X_{k}]=[Y_{j},Y_{k}]=0\quad\mbox{and}\quad[X_{j},Y_{k}]=\delta_{jk}Z.$

For $\xi\in\Lambda$, a short computation shows that

$\mathfrak{h}_{V}(\xi)=V_{0}\oplus\mathbb{R}.$

We conclude that

$\mathcal{H}_{V}=(\mathfrak{z}^{*}\setminus\\{0\\})\times L^{2}(JV_{0}),$

and the $G$-representation in each fiber is given by

$\pi_{\xi}(X+JY+tZ)f(x):=\mathrm{e}^{i\xi(t+\omega(X,x))}f(x+JY),\quad X,Y\in
V_{0},\;t\in\mathbb{R},\;f\in L^{2}(JV_{0}),$

where we identify $\mathfrak{z}^{*}=\mathbb{R}$ via the basis $Z$. The reader
should note that the construction does not depend on the Jordan-Hölder basis
in this case, but only on the complex structure and the Lagrangian subspace
$V_{0}$. This gives an explicit isomorphism $I_{G}\cong
C_{0}(\mathfrak{z}^{*}\setminus\\{0\\},\mathbb{K}(L^{2}(JV_{0})))$ and a short
exact sequence

$0\to C_{0}(\mathfrak{z}^{*}\setminus\\{0\\},\mathbb{K}(L^{2}(JV_{0})))\to
C^{*}(\mathbb{H}_{n})\to C_{0}(V^{*})\to 0.$

We also note that since $\Gamma$ consists of one fine strata, Proposition
0.8.14 implies that $[\zeta]=0\in
H^{1}(\mathrm{Aut}(\mathbb{H}_{n}),\check{H}^{1}(\Gamma,U(1)))$ (which also
follows from that
$\check{H}^{1}(\Gamma,U(1))=\check{H}^{1}(\mathbb{R}^{\times},U(1))\cong 0$.

It is now possible to describe the cocycle $c_{\mathrm{Aut}(\mathbb{H}_{n})}$.
Recall the definition of the flatness cocycle
$c_{\mathrm{Aut}(\mathbb{H}_{n})}\in Z^{2}(\mathrm{Aut}(\mathbb{H}_{n}),U(1))$
from Definition 0.8.9.

######  Proposition 0.9.12.

Let $\mathbb{H}_{n}$ denote the Heisenberg Lie group constructed from
$V\ltimes_{\omega}\mathbb{R}$. The cohomology class
$[c_{\mathrm{Aut}(\mathbb{H}_{n})}]\in
H^{2}(\mathrm{Aut}(\mathbb{H}_{n}),U(1))$ is non-trivial and belongs to the
image of

$H^{2}(\mathrm{Aut}(\mathbb{H}_{n}),C(\Gamma,\mathbb{Z}/8))\to
H^{2}(\mathrm{Aut}(\mathbb{H}_{n}),C(\Gamma,U(1))).$

######  Remark 0.9.13.

The action of ${\rm Sp}(V,\omega)\subseteq\mathrm{Aut}(\mathbb{H}_{n})$ on
$\Gamma$ is trivial. It is well known that ${\rm Sp}(V,\omega)$ has a unique
two fold cover $\widetilde{{\rm Sp}}(V,\omega)$ and the action on $I$ lifts to
an action of $\widetilde{{\rm Sp}}(V,\omega)$ on
$\mathcal{H}_{V}=(\mathfrak{z}^{*}\setminus\\{0\\})\times L^{2}(iV_{0})$ – the
Segal-Shale-Weil representation. See more in [71, Chapter 4]. In particular,
$[c_{{\rm Sp}(V,\omega)}]$ is in the image of $H^{2}({\rm
Sp}(V,\omega),\mathbb{Z}/2)\to H^{2}({\rm Sp}(V,\omega),U(1))$.

######  Proposition 0.9.14.

Let $\mathbb{H}_{n}$ denote the Heisenberg Lie group constructed from
$V\ltimes_{\omega}\mathbb{R}$. The action of $\widetilde{{\rm Sp}}(V,\omega)$
on $\mathcal{H}_{V}$ extends uniquely to an action of $\widetilde{{\rm
Sp}}(V,\omega)\rtimes\mathbb{Z}/2$ lifting the action on $I_{\mathbb{H}_{n}}$.
In particular, for

$\text{\mxedr.g}=(SO(V)\cap
Sp(V,\omega))\rtimes\mathbb{Z}_{2}=U(V)\rtimes\mathbb{Z}_{2},$

we have that

$[c_{\text{\mxedr.g}}]=0\in H^{2}(\text{\mxedr.g},C(\Gamma,U(1))).$

In particular, there is a .g-equivariant isomorphism $I_{G}\cong
C_{0}(\mathfrak{z}^{*}\setminus\\{0\\},\mathbb{K}(L^{2}(JV_{0})))$ for a
suitable unitary representation of .g on the trivial bundle
$\mathfrak{z}^{*}\setminus\\{0\\}\times
L^{2}(JV_{0})\to\mathfrak{z}^{*}\setminus\\{0\\}$.

###### Proof.

We can also lift the action of ${\rm
Sp}(V,\omega)\rtimes\mathbb{Z}/2\subseteq\mathrm{Aut}(\mathbb{H}_{n})$ to a
projective action on $\mathcal{H}_{V}$. We can decompose $\mathcal{H}_{V}$
over
$\mathfrak{z}^{*}\setminus\\{0\\}=\mathbb{R}_{<0}\dot{\cup}\mathbb{R}_{>0}$ as
$\mathcal{H}_{V}=\mathcal{H}_{V<}\dot{\cup}\mathcal{H}_{V>0}$. The
$\mathbb{Z}/2$-factor interchanges the two pieces of $\mathcal{H}_{V}$ and we
can lift its action to the fiber by defining it as complex conjugation
$\mathcal{H}_{V,\xi}=L^{2}(JV_{0})\to L^{2}(JV_{0})=\mathcal{H}_{V,-\xi}$.
This produces an action of $\mathbb{Z}/2$ on $\mathcal{H}_{V}$ that lifts the
action on $I_{\mathbb{H}_{n}}$. A short comparison with the construction of
the Segal-Shale-Weil representation (see [71, Equation (4.24), (4.25),
(4.26)]) shows that this induces an action of $\widetilde{{\rm
Sp}}(V,\omega)\rtimes\mathbb{Z}/2$.

The final conclusion of the proposition follows from that $\widetilde{{\rm
Sp}}(V,\omega)$ is defined from a cocycle which is cohomologically trivial on
$U(V)$ (as the Weil-Shale-Weil representation restricts to a representation of
$U(V)$), so the inclusion $U(V)\to{\rm Sp}(V,\omega)$ factors over an
inclusion $U(V)\to\widetilde{{\rm Sp}}(V,\omega)$. ∎

######  Example 0.9.15.

Let $\mathbb{H}_{n,\mathbb{C}}$ be the complexification of the Heisenberg
group. That is we define its Lie algebra as
$\mathfrak{h}_{n,\mathbb{C}}:=\mathfrak{h}_{n}\otimes_{\mathbb{R}}\mathbb{C}$
viewed as a real Lie algebra. The complex Heisenberg group is the local model
for certain polycontact structures constructed from complex manifolds, see
Example 0.16.7 below. If $\mathfrak{h}_{n}$ is defined from an inner product
space $V$ with a non-degenerate $2$-form $\omega$, then identifying
$V_{\mathbb{C}}:=V\otimes_{\mathbb{R}}\mathbb{C}=V+iV=V\oplus V$, we have that
$\mathfrak{h}_{n,\mathbb{C}}=(V\oplus V)_{W}$ where
$W\subseteq\mathfrak{so}(V\oplus V)$ is spanned by

$\omega^{1}=\begin{pmatrix}\omega&0\\\
0&-\omega\end{pmatrix}\quad\mbox{and}\quad\omega^{2}=\begin{pmatrix}0&\omega\\\
\omega&0\end{pmatrix}.$

Note that the complexification $\omega_{\mathbb{C}}$ of $\omega$ can be
identified with $\omega^{1}+i\omega^{2}$. The constructions can be carried out
starting from a complex vector space $V_{\mathbb{C}}$ equipped with a complex
non-degenerate $2$-form $\omega_{\mathbb{C}}$, but for simplicity we start
from $V$ which produces a real subspace of $V_{\mathbb{C}}$ which is a
Lagrangian for the imaginary part of $\omega_{\mathbb{C}}$.

The Lie algebra
$\mathfrak{h}_{n,\mathbb{C}}:=\mathfrak{h}_{n}\otimes_{\mathbb{R}}\mathbb{C}$
is a step $2$ nilpotent Lie algebra of type $(2,2\dim(V))$ with
$C(\mathfrak{h}_{n,\mathbb{C}})=\mathfrak{z}$. Most things are quite similar
to the Heisenberg group reviewed above in Example 0.9.11,. For instance, there
is a standard grading defined from
$(\mathfrak{h}_{n,\mathbb{C}})_{-1}=V\otimes_{\mathbb{R}}\mathbb{C}$.
Moreover, we have that $\mathrm{Aut}_{\rm
gr}(\mathbb{H}_{n,\mathbb{C}})=u_{H}(H_{\mathfrak{h}_{n,\mathbb{C}}})$, where
$H_{\mathfrak{h}_{n,\mathbb{C}}}\subseteq GL_{\mathbb{R}}(V\otimes\mathbb{C})$
consists of those $g$ preserving the linear span of $\omega^{1}$ and
$\omega^{2}$. The group $H_{\mathfrak{h}_{n,\mathbb{C}}}$ contains a subgroup
of the form
$\mathrm{Sp}_{\mathbb{C}}(V_{\mathbb{C}},\omega_{\mathbb{C}})\times
U(1)\times\mathbb{R}_{>0}$, where
$\mathrm{Sp}_{\mathbb{C}}(V_{\mathbb{C}},\omega_{\mathbb{C}})$ denotes the
complex symplectic group on the complex symplectic space
$(V_{\mathbb{C}},\omega_{\mathbb{C}})$, and $U(1)$ acts via complex
multiplication on $V_{\mathbb{C}}$. We also have that

$\mathrm{Aut}(\mathbb{H}_{n,\mathbb{C}})\cong\mathrm{Hom}_{\mathbb{R}}(V_{\mathbb{C}},\mathbb{R}^{2})\rtimes
H_{\mathfrak{h}_{n,\mathbb{C}}}.$

For $\xi=(\xi_{1},\xi_{2})\in\mathfrak{z}^{*}$, a short computation proves

$\mathsf{Pf}(\omega_{\xi})=\mathsf{Pf}(\xi_{1}\omega^{1}+\xi_{2}\omega^{2})=(\xi_{1}^{2}+\xi_{2}^{2})^{\dim(V)/2}.$

Therefore, the flat orbits are computed as

$\Gamma=\mathfrak{z}^{*}\setminus\\{0\\}\quad\mbox{and}\quad\Gamma_{\partial}\cong
S^{1}.$

The action of $\mathrm{Aut}(\mathbb{H}_{n,\mathbb{C}})$ on $\Gamma$ is linear,
so the action on $\Gamma_{\partial}$ factors over a quotient
$\mathrm{Aut}(\mathbb{H}_{n,\mathbb{C}})\to U(1)$ acting by rotation. A big
difference to the real Heisenberg group is that $\Gamma$ consists of multiple
fine stratas; from a choice of ON-basis on $V$ we can construct a Jordan-
Hölder basis of $\mathfrak{h}_{n,\mathbb{C}}$ and the top fine strata in this
basis will be $\Gamma$ with a coordinate axis removed.

Following the same arguments as in Example 0.9.11, for any Lagrangian
$V_{0}\subseteq V$, the abelian Lie algebra $V_{0,\mathbb{C}}+\mathfrak{z}$
defines a polarization of all flat orbits, and the representation defined from
$\xi\in\Gamma$ takes the form

$\pi_{\xi}(X+JY+tZ)f(x):=\mathrm{e}^{i\xi(t+\omega_{\mathbb{C}}(X,x))}f(x+JY),$

for $X,Y\in V_{0,\mathbb{C}}$, $t\in\mathbb{C}$, and $f\in
L^{2}(JV_{0,\mathbb{C}})$. Here we have fixed a complex structure $J$ on $V$,
and where we have identified $\mathfrak{z}=\mathbb{C}$ and $\xi$ with an
$\mathbb{R}$-linear functional $\mathbb{C}\to\mathbb{R}$. Similarly to Example
0.9.11, we arrive at an explicit isomorphism
$I_{\mathbb{H}_{n,\mathbb{C}}}\cong
C_{0}(\mathfrak{z}^{*}\setminus\\{0\\},\mathbb{K}(L^{2}(JV_{0,\mathbb{C}})))$
and a short exact sequence

$0\to
C_{0}(\mathfrak{z}^{*}\setminus\\{0\\},\mathbb{K}(L^{2}(JV_{0,\mathbb{C}})))\to
C^{*}(\mathbb{H}_{n,\mathbb{C}})\to C_{0}(V^{*}_{\mathbb{C}})\to 0.$

Since there is a global polarization of the flat orbits, Proposition 0.8.14
implies that $\delta_{\rm DD}(I_{\mathbb{H}_{n,\mathbb{C}}})=0\in
H^{3}(\mathfrak{z}^{*}\setminus\\{0\\},\mathbb{Z})$ (which also follows from
the argument above) and $[\zeta]=0\in
H^{1}(\mathrm{Aut}(\mathbb{H}_{n,\mathbb{C}}),\check{H}^{1}(\mathfrak{z}^{*}\setminus\\{0\\},U(1)))=H^{1}(\mathrm{Aut}(\mathbb{H}_{n,\mathbb{C}}),\mathbb{Z})$.
We moreover have that

$[c_{\mathrm{Sp}_{\mathbb{C}}(V_{\mathbb{C}},\omega_{\mathbb{C}})}]=0,$

since the $\mathrm{Sp}_{\mathbb{C}}(V_{\mathbb{C}},\omega_{\mathbb{C}})$ is
simply connected. In particular, we conclude from Theorem 0.8.11 that there is
an $\mathrm{Sp}_{\mathbb{C}}(V_{\mathbb{C}},\omega_{\mathbb{C}})$-equivariant
isomorphism

$I_{\mathbb{H}_{n,\mathbb{C}}}\cong
C_{0}(\mathfrak{z}^{*}\setminus\\{0\\},\mathbb{K}(L^{2}(JV_{0,\mathbb{C}}))),$

for a suitable unitary representation of
$\mathrm{Sp}_{\mathbb{C}}(V_{\mathbb{C}},\omega_{\mathbb{C}})$ on the trivial
bundle $\mathfrak{z}^{*}\setminus\\{0\\}\times
L^{2}(JV_{0,\mathbb{C}})\to\mathfrak{z}^{*}\setminus\\{0\\}$. We note that
$\mathrm{Sp}_{\mathbb{C}}(V_{\mathbb{C}},\omega_{\mathbb{C}})$ acts trivially
on $\Gamma=\mathfrak{z}^{*}\setminus\\{0\\}$ since it acts trivially on the
center of the Lie algebra.

######  Example 0.9.16.

Let $\mathsf{G}$ be a product of Heisenberg groups. This is the local model
for the tangent group of a product of contact manifolds. For notational
convenience we write
$\mathsf{G}=\mathbb{H}_{n_{1}}\times\ldots\times\mathbb{H}_{n_{k}}$ and assume
that each $\mathbb{H}_{n_{j}}$ is defined from a non-zero real inner product
space $V_{j}$ with equipped with a non-degenerate
$\omega^{j}\in\mathfrak{so}(V_{j})$. Then $\mathsf{G}$ is of type $(p,q)$
where:

$p=k,\text{ and }q=2(n_{1}+\ldots n_{k}).$

For notational convenience, we write $V:=\oplus_{j=1}^{k}V_{j}$ and
$\mathfrak{z}=\oplus_{j=1}^{k}\mathfrak{z}_{j}$ where $\mathfrak{z}_{j}$ is
the center of the $j$:th factor. We note that $C(\mathfrak{g})=\mathfrak{z}$
in this case. We call the grading

$\mathfrak{g}_{-1}:=V\quad\mbox{and}\quad\mathfrak{g}_{-1}:=\mathfrak{z},$

the standard grading on $\mathsf{G}$ as it it is the only grading for which
all projections onto the standardly graded factors $\mathbb{H}_{n_{j}}$ are
graded.

Let us describe the automorphisms. By Proposition 0.9.3,

$\mathrm{Aut}(\mathsf{G})\cong\mathrm{Hom}(V,\mathfrak{z})\rtimes
H_{\mathfrak{g}},$

where we for rank reasons can write

$\displaystyle H_{\mathfrak{g}}$ $\displaystyle=\\{\varphi\in
GL(V):\exists(\lambda_{ij}(\varphi))_{i,j=1}^{k}\mbox{ s.t.
}\;\varphi^{t}\omega_{i}\varphi=\sum_{j=1}^{k}\lambda_{ij}(\varphi)\omega_{j}\\}=$
$\displaystyle=\\{\varphi\in GL(V):\forall i\exists j,\lambda\neq 0\mbox{ with
}n_{i}=n_{j}\mbox{ s.t.
}\;\varphi^{t}\omega_{i}\varphi=\lambda\omega_{j}\\}\cong$
$\displaystyle\cong(\mathbb{R}_{>0}^{k}\times({\rm
Sp}(V_{1},\omega_{1})\times\cdots\times{\rm
Sp}(V_{k},\omega_{k}))\rtimes(\mathbb{Z}/2)^{k})\rtimes
F_{(n_{1},\ldots,n_{k})},$

and $F_{(n_{1},\ldots,n_{k})}$ is the finite group of all permutations on $k$
elements $\sigma\in S_{k}$ such that $n_{i}=n_{\sigma(i)}$ for all
$i=1,\ldots,k$. If $\mathsf{G}$ is equipped with the standard grading, then

$\mathrm{Aut}_{\rm gr}(\mathsf{G})=u_{H}(H_{\mathfrak{g}}).$

The flat orbits are readily described because we are dealing with a product. A
similar argument as in Example 0.9.11 shows that

$\Gamma=(\mathfrak{z}^{*}_{1}\setminus\\{0\\})\times\cdots\times(\mathfrak{z}^{*}_{k}\setminus\\{0\\})=\\{\xi=(\xi_{1},\ldots,\xi_{k})\in\mathfrak{z}^{*}:\xi_{1}\cdots\xi_{k}\neq
0\\}.$

A computation shows that $\Gamma$ consists of one fine strata in a suitable
Jordan-Hölder basis. The action of $\mathrm{Aut}(\mathsf{G})$ on $\Gamma$
factors over the quotient mapping

$\mathrm{Aut}(\mathsf{G})\to(\mathbb{R}_{>0}^{k}\times(\mathbb{Z}/2)^{k})\rtimes
F_{(n_{1},\ldots,n_{k})}=(\mathbb{R}^{\times})^{k}\rtimes
F_{(n_{1},\ldots,n_{k})},$

and $(\mathbb{R}^{\times})^{k}\rtimes F_{(n_{1},\ldots,n_{k})}$ acts on
$\Gamma$ by rescaling and permutation of the coordinates. By Proposition
0.7.6, we can identify

$\Gamma_{\partial}=\\{\xi=(\xi_{1},\ldots,\xi_{k})\in\mathfrak{z}^{*}:\xi_{1}\cdots\xi_{k}=\pm
1\\}.$

We remark that if we choose a different grading than the standard grading, the
action of $\mathrm{Aut}_{\rm gr}(\mathsf{G})$ on $\Gamma_{\partial}$ can be
quite complicated.

If we pick an adapted complex structure on $V_{j}$ and a Lagrangian subspaces
$V_{0,j}\subseteq V_{j}$ for each $j$, an argument as in Example 0.9.11 shows
that we for a suitable Jordan-Hölder basis arrive at the Vergne polarization

$\mathfrak{h}_{V}(\xi)=V_{0}\oplus\mathfrak{z},$

for each $\xi\in\Gamma$ where $V_{0}:=\bigoplus_{j=1}^{k}V_{0,j}$. We conclude
that

$\mathcal{H}_{V}=\Gamma\times L^{2}(JV_{0}),$

and the $\mathsf{G}$-representation in the fiber over
$\xi\in\Gamma\subseteq\mathfrak{z}^{*}$ is given by

$\pi_{\xi}(X+JY+Z)f(x):=\mathrm{e}^{i\sum_{j=1}^{k}\xi_{j}(Z_{j}+\omega^{j}(X_{j},x_{j}))}f(x+JY),$

for $X,Y\in V_{0}$, $Z\in\mathfrak{z}$, and $f\in L^{2}(JV_{0})$.

Following the argument for the Heisenberg group from Proposition 0.9.14, we
can describe the restriction of the flatness cocycle $c_{\rm flat}\in
Z^{2}(\mathrm{Aut}(\mathsf{G}),C(\Gamma,\mathbb{C}^{\times}))$ to certain
subgroups. We remark that Proposition 0.9.14 ensures that $[c_{\rm flat}]\in
H^{2}(\mathrm{Aut}(\mathsf{G}),C(\Gamma,\mathbb{C}^{\times}))$ is non-trivial
as its restriction to $H^{2}({\rm Sp}(V_{1},\omega_{1})\times\cdots\times{\rm
Sp}(V_{k},\omega_{k}),C(\Gamma,\mathbb{C}^{\times}))$ is non-trivial (but
$\mathbb{Z}/2$-valued). We note that for

$\text{\mxedr.g}_{0}:=(U(V_{1})\times\cdots\times
U(V_{k}))\rtimes(\mathbb{Z}/2)^{k},$

and an argument as in Proposition 0.9.14 implies that
$[c_{\text{\mxedr.g}_{0}}]=0\in
H^{2}(\text{\mxedr.g}_{0},C(\Gamma,\mathbb{C}^{\times}))$. Moreover, for

$\text{\mxedr.g}:=\text{\mxedr.g}_{0}\rtimes F_{(n_{1},\ldots,n_{k})}$

we have by a standard argument that

$[c_{\text{\mxedr.g}}]\in\mathrm{i}\mathrm{m}\,(H^{2}(F_{(n_{1},\ldots,n_{k})},C(\Gamma,\mathbb{C}^{\times}))\to
H^{2}(\text{\mxedr.g},C(\Gamma,\mathbb{C}^{\times})).$

######  Example 0.9.17.

For $n>1$, we consider the $2n-1$-dimensional Lie algebra $\mathfrak{g}$
spanned by $X_{1},\ldots,X_{n},Y_{1},\ldots,Y_{n-1}$ such that

$[X_{i},X_{i+1}]=Y_{i},\quad i=1,2,\ldots,n-1$

with all other brackets in the basis vectors being zero. The Lie algebra
$\mathfrak{g}$ is nilpotent of step length $2$ and arises as the quotient of
the nilpotent Lie algebra of step length $n$ consisting of all real
$(n+1)\times(n+1)$ upper triangular nilpotent matrices (see more in Example
0.9.23) by the ideal of all its $n-2$:nd order commutators. We have that
$C(\mathfrak{g})=\mathfrak{z}=\oplus_{j=1}^{n-1}\mathbb{R}Y_{j}$, and
$\mathfrak{g}$ is of type $(n-1,n)$ and is determined by
$\omega^{1},\ldots,\omega^{n-1}\in\mathfrak{so}(n)$ given as

$\omega^{j}=\iota_{j}\omega^{0}q_{j},$

where $\omega^{0}$ is the standard symplectic form on $\mathbb{R}^{2}$,
$\iota_{j}:\mathbb{R}^{2}\to\mathbb{R}^{n}$ is the embedding in the oriented
basis $(X_{j},X_{j+1})$ and $q_{j}:=\iota_{j}^{*}$ is the projection
$\mathbb{R}^{n}\to\mathbb{R}^{2}$ onto the basis $(X_{j},X_{j+1})$. We note
that the associated linear span $W\subseteq\mathfrak{so}(n)$ consists of all
anti-symmetric matrices $\omega=(\omega_{ij})_{i,j=1}^{n}$ with
$\omega_{ij}=0$ if $|i-j|>1$.

A grading on $\mathfrak{g}$ is characterized by an $n$-tuple
$(l_{1},\ldots,l_{n})$, where $l_{j}\geq 1$ for all $j$ with equality for at
least one $j$. The grading is defined so that $X_{j}$ has degree $l_{j}$.
Indeed, for such an $n$-tuple we can write

$\mathfrak{g}=\bigoplus_{p<0}\mathfrak{g}_{p},\quad\mbox{where}\quad\mathfrak{g}_{p}:=\left(\bigoplus_{j:\,l_{j}=p}\mathbb{R}X_{j}\right)\oplus\left(\bigoplus_{j:\,l_{j}+l_{j+1}=p}\mathbb{R}Y_{j}\right).$

We call $(l_{1},\ldots,l_{k})=(1,\ldots,1)$ the standard grading on
$\mathfrak{g}$, in this grading the center has degree $2$.

The automorphism group of $\mathfrak{g}$ is determined by Proposition 0.9.3,
so

$\mathrm{Aut}(\mathfrak{g})\cong\mathrm{Hom}(\mathbb{R}^{n},\mathbb{R}^{n-1})\rtimes
H_{\mathfrak{g}},$

where

$H_{\mathfrak{g}}:=\\{g\in
GL_{n}(\mathbb{R}):(g^{t}\omega^{i}g)_{jk}=0\;\forall
i=1,\ldots,n-1,\;|j-k|>1\\}.$

In general, there are very few compact subgroups of $H_{\mathfrak{g}}$. In the
standard grading, $\mathrm{Aut}_{\rm
gr}(\mathfrak{g})=u_{H}(H_{\mathfrak{g}})$.

######  Proposition 0.9.18.

Consider the Lie algebra $\mathfrak{g}$ from the preceding paragraphs with
$n=4$, i.e. the $2$-step $7$-dimensional Lie algebra defined from the bracket
relations

$[X_{1},X_{2}]=Y_{1},\;[X_{2},X_{3}]=Y_{2},\;[X_{3},X_{4}]=Y_{3}.$

Let $V$ denote the linear span of $\\{X_{1},X_{2},X_{3},X_{4}\\}$. Then for
any Riemannian metric on $V$, $H_{\mathfrak{g}}\cap O(V)$ is finite. In
particular, there are no infinite compact subgroups of $\mathrm{Aut}_{\rm
gr}(\mathfrak{g})$ for $n=4$.

The take away from this proposition is that in order to expect interesting
geometric examples modelled on the Lie algebra appearing in this example, we
need to use non-compact structure groups.

###### Proof.

We have that $H_{\mathfrak{g}}\cap O(V)$ is compact, so it suffices to prove
that $(H_{\mathfrak{g}})_{(0)}\cap O(V)$ is finite, where
$(H_{\mathfrak{g}})_{(0)}$ denotes the connected component containing the
identity. A lengthier computation with the Lie algebra of $H_{\mathfrak{g}}$
shows that

$(H_{\mathfrak{g}})_{(0)}=\left\\{g\in GL(4):g=\begin{pmatrix}g_{11}&0&0&0\\\
g_{21}&g_{22}&0&g_{24}\\\ g_{31}&0&g_{33}&g_{34}\\\
0&0&0&g_{44}\end{pmatrix}\right\\}.$

For $g\in(H_{\mathfrak{g}})_{(0)}$ we have that $g\in O(V)$ if and only if

$\begin{cases}g_{22}=\pm 1\\\ g_{33}=\pm 1\\\
g_{21}g_{22}=g_{22}g_{24}=g_{24}g_{22}=g_{34}g_{33}=0\\\
g_{11}^{2}+g_{21}^{2}+g_{31}^{2}=1,\\\ g_{24}^{2}+g_{34}^{2}+g_{44}^{2}=1,\\\
g_{21}g_{24}+g_{31}g_{34}=0.\end{cases}$

The first three equations implies that $g\in(H_{\mathfrak{g}})_{(0)}\cap O(V)$
if and only if $g\in O(V)$ is diagonal. We conclude that
$(H_{\mathfrak{g}})_{(0)}\cap O(V)\cong(\mathbb{Z}/2)^{4}$ is finite.

The final conclusion follows from that if
$\text{\mxedr.g}\subseteq\mathrm{Aut}_{\rm
gr}(\mathfrak{g})=u_{H}(H_{\mathfrak{g}})$ is compact it acts isometrically on
$V$ for some Riemannian metric and by the previous argument we have that .g is
finite. ∎

We compute the flat orbits using Equation (19):

$\displaystyle\Gamma$
$\displaystyle=\\{\xi=(\xi_{1},\ldots,\xi_{n-1})\in\mathfrak{z}^{*}:\det(\xi_{1}\omega^{1}+\cdots+\xi_{n-1}\omega^{n-1})\neq
0\\}=$ $\displaystyle=\begin{cases}\emptyset,\;&\mbox{if $n$ is odd},\\\
\left\\{\xi=(\xi_{1},\ldots,\xi_{n-1})\in\mathfrak{z}^{*}:\xi_{1}\xi_{3}\cdots\xi_{n-1}\neq
0\right\\},\;&\mbox{if $n$ is even}.\end{cases}$

The space $\Gamma$ consists of multiple fine stratas. Indeed, for $n$ odd, any
antisymmetric matrix is degenerate and for $n$ even induction on $n/2$ shows
that
$\det(\xi_{1}\omega^{1}+\cdots+\xi_{n-1}\omega^{n-1})=\xi_{1}^{2}\xi_{3}^{2}\cdots\xi_{n-1}^{2}$.
Using Proposition 0.7.6, we identify

$\Gamma_{\partial}=\begin{cases}\emptyset,\;&\mbox{if $n$ is odd},\\\
\left\\{\xi=(\xi_{1},\ldots,\xi_{n-1})\in\mathfrak{z}^{*}:\xi_{1}\xi_{3}\cdots\xi_{n-1}=\pm
1\right\\},\;&\mbox{if $n$ is even}.\end{cases}$

######  Remark 0.9.19.

Consider $n=3$ for which the above construction produces the 5-dimensional Lie
algebra $\mathfrak{g}$ spanned by $X_{1},X_{2},X_{3},Y_{1},Y_{2}$ subject to
the relations

$[X_{1},X_{2}]=Y_{1},\ [X_{2},X_{3}]=Y_{2},$

and all other brackets being zero. This nilpotent Lie algebra $\mathfrak{g}$
is of step length $2$, of type $(2,3)$ with $C(\mathfrak{g})=\mathfrak{z}$ but
admits no flat orbits.

We henceforth restrict our attention to $n$ being even, so that there exists
flat orbits. The Vergne polarization of the flat orbits is computed using
Proposition 0.6.8 to be

$\mathfrak{h}_{V}(\xi)=\mathfrak{h}=\mathfrak{z}+\sum_{j=1}^{n/2}\mathbb{R}X_{2j-1},$

for all $\xi\in\Lambda$. Note that
$\mathfrak{h}_{V}(\xi)=\mathfrak{h}\subseteq\mathfrak{g}$ is abelian and
independent of $\xi$. Using the same argument as in Proposition 0.9.9 we
conclude that

$\mathcal{H}_{V}=\Lambda\times L^{2}(\mathbb{R}^{n/2}),$

and the $\mathsf{G}$-representation in the fiber over
$\xi\in\Gamma\subseteq\mathfrak{z}^{*}$ is given by

$\displaystyle\pi_{\xi}$
$\displaystyle\left(\mathrm{exp}\left(\sum_{j=1}^{n}x_{j}X_{j}+\sum_{l=1}^{n-1}y_{j}Y_{j}\right)\right)f(t_{1},t_{3},\ldots,t_{n-1}):=$
$\displaystyle:=\mathrm{e}^{ih(\xi,t,x,y)}f(t_{1}+x_{1},t_{3}+x_{3},\ldots,t_{n-1}+x_{n-1}),$

for $f\in L^{2}(\mathbb{R}^{n/2})$, where

$\displaystyle h(\xi,t,x,y)$
$\displaystyle=\xi\left(\sum_{l=1}^{n-1}y_{j}Y_{j}\right)+\xi\left[\sum_{j=1}^{n}x_{j}X_{j},\sum_{l=1}^{n/2}t_{2l-1}X_{2l-1}\right]=$
$\displaystyle=\sum_{j=1}^{n-1}\xi_{j}y_{j}-\xi_{1}x_{1}t_{1}+\sum_{l=2}^{n/2}(\xi_{2l-2}x_{2l-2}-\xi_{2l-1}x_{2l})t_{2l-1}.$

Since there is a global polarization of the flat orbits, Proposition 0.8.14
implies that $[\zeta]=0\in
H^{1}(\mathrm{Aut}(\mathsf{G}),\check{H}^{1}(\Gamma,U(1)))$. In lack of a good
description of the automorphisms, we settle for noting that there is an
$\mathrm{Aut}(\mathsf{G})$-equivariant isomorphism $I_{\mathsf{G}}\cong
C_{0}(\Gamma,\mathbb{K}(L^{2}(\mathbb{R}^{n/2})))$ for some projective
$\mathrm{Aut}(\mathsf{G})$-action on the trivial bundle $\Gamma\times
L^{2}(\mathbb{R}^{n/2})\to\Gamma$.

######  Example 0.9.20.

Consider the free step $2$ nilpotent Lie group $\mathsf{G}$ on a finite-
dimensional real inner product space $V$. In the notations of above,
$\mathfrak{g}=V_{\mathfrak{so}(V)}$ and $\mathfrak{g}$ satisfies
$C(\mathfrak{g})=\mathfrak{z}=\mathfrak{so}(V)$. Indeed, the only nontrivial
bracket is for $x,y\in V\subseteq\mathfrak{g}$ and it is given by

$[x,y]=x\wedge y,$

under the identification $\mathfrak{so}(V)\cong\wedge^{2}V$ defined from the
inner product. The free step $2$ nilpotent Lie algebra is of type
$(n(n-1)/2,n)$ where $n:=\dim(V)$.

There is a one-to-one correspondence of vector space gradings
$V=\bigoplus_{p<0}V_{p}$ with $V_{-1}\neq 0$ and Lie algebra gradings
$\mathfrak{g}=\bigoplus_{p<0}\mathfrak{g}_{p}$ given by associating a grading
on $\mathfrak{g}$ with a grading on $V$ by

$\mathfrak{g}_{p}:=V_{p}\oplus\left(\bigoplus_{j+l=p}V_{j}\wedge
V_{l}\right).$

The inverse construction induces a grading on $V$ from a grading on
$\mathfrak{g}$ by declaring the quotient $\mathfrak{g}\to V$ to be graded. We
call the grading with $V$ having constant degree $-1$ the standard grading.

By universality and Proposition 0.9.3,

$\mathrm{Aut}(\mathsf{G})\cong\mathrm{Hom}(V,\mathfrak{so}(V))\rtimes GL(V),$

and

$\mathrm{Aut}_{\rm gr}(\mathsf{G})=u_{H}(GL_{\rm gr}(V)),$

in the standard grading. Equation (19) shows that we can identify

$\Gamma=\mathfrak{so}(V)\cap
GL(V)\quad\mbox{and}\quad\Gamma_{\partial}=\mathfrak{so}(V)\cap SL(V),$

using the self-duality $\mathfrak{so}(V)^{*}=\mathfrak{so}(V)$. We note that
$\Gamma$ is empty if $\dim(V)$ is odd and in this case readily seen to be a
Zariski open subset of $\mathfrak{so}(V)$ if $\dim(V)$ is even. The action of
the automorphisms of $\mathsf{G}$ factors over the surjection
$\mathrm{Aut}(\mathsf{G})\to GL(V)$ and $GL(V)$ acts on
$\Gamma=\mathfrak{so}(V)\cap GL(V)$ by matrix multiplication

$\varphi.\xi:=\varphi^{t}\xi\varphi\in\mathfrak{so}(V)\cap GL(V).$

Since any non-degenerate two form admits a basis in which it is the standard
symplectic form $\omega_{0}$ on $V$, the $GL(V)$-action is transitive and
there is a $GL(V)$-equivariant homeomorphism

$GL(V)/\mathrm{Sp}(V,\omega_{0})\xrightarrow{\sim}\Gamma.$

As above, we pick a Jordan-Hölder basis for $\mathfrak{g}$ with
$\mathfrak{g}_{n(n-1)/2}=\mathfrak{z}=\mathfrak{so}(V)$ (for $n=\dim(V)$).
Such a Jordan-Hölder basis induces a basis of $V$ and a filtration
$V_{j}:=\mathfrak{g}_{n(n-1)/2+j)}/\mathfrak{z}$ of $V$. We can describe the
Vergne polarization of the flat orbits using Proposition 0.6.8 as

$\mathfrak{h}_{V}(\xi)=\mathfrak{z}+\sum_{j=1}^{\dim(V)}\ker(\xi|_{V_{j}}),$

for all $\xi\in\Lambda=\mathfrak{so}(V)\cap GL(V)$. We note that
$\mathfrak{h}_{V}(\xi)\subseteq\mathfrak{g}$ is neither abelian nor constant
in $\xi\in\Lambda$. Moreover, it does not extend from the top fine strata
(defined relative to the fixed Jordan-Hölder basis) to all flat orbits. We
shall therefore take a more direct route using Fock-Bargmann spaces (see for
instance [71, Chapter 1.6]) to constructing a bundle of flat representations.
An additional feature that comes in handy is that orthogonal automorphisms act
most naturally on the Fock-Bargmann spaces. We only consider the case that $V$
is even-dimensional, so there exists flat orbits.

We fix an inner product metric on $V$, it sets up a linear bijection between
antisymmetric forms and $\mathfrak{so}(V)\subseteq\mathrm{End}(V)$, and
between symmetric forms and symmetric elements of $\mathrm{End}(V)$. In terms
of this fixed inner product, there is a bijective correspondences between
symplectic forms $\omega$ and pairs $(J,g)$ of inner products $g$ on $V$ and a
complex structure $J$ on $V$ with $g(x,Jy)=-g(Jx,y)$ for all $x,y\in V$.
Indeed, the pair $(J,g)$ determines the symplectic form
$\omega_{(J,g)}(x,y)=g(x,Jy)$, and conversely we can given $\omega$ define
$g_{\omega}:=(-\omega^{2})^{1/2}$ and
$J_{\omega}:=(-\omega^{2})^{-1/2}\omega$. We remark that

$\langle x,y\rangle_{\omega}:=g_{\omega}(x,y)+i\omega(x,y),$

defines a hermitean inner product on $V$.

Given a symplectic form $\omega$ on $V$, we can define the associated Fock-
Bargmann space as $\mathpzc{F}(V,\omega):=\mathcal{O}(V)\cap
L^{2}(V,\mathrm{e}^{-g_{\omega}})$. That is, $\mathpzc{F}(V,\omega)$ is the
subspace of holomorphic functions (with respect to the complex structure
$J_{\omega}$) in the weighted $L^{2}$-space
$L^{2}(V,\mathrm{e}^{-g_{\omega}})$ defined from the weight
$\mathrm{e}^{-g_{\omega}}(x):=\mathrm{e}^{-g_{\omega}(x,x)}$. One can
equivalently define
$\mathpzc{F}(V,\omega):=\bigoplus_{k=0}^{\infty}V^{*\otimes_{\mathbb{C}}^{s}k}$,
were $V^{*\otimes_{\mathbb{C}}^{s}0}=\mathbb{C}$ and
$V^{*\otimes_{\mathbb{C}}^{s}k}$ is the $k$-fold symmetric tensor product of
$V^{*}$ over $\mathbb{C}$ defined with respect to the complex structure
$J_{\omega}$, with Hilbert space structure defined from $g_{\omega}$.

The Bargmann-Fock representation of
$\mathfrak{h}_{\dim(V)/2}=V_{\mathbb{R}\omega}$ (see for instance [71, Chapter
1.6]) extends to the free step $2$ nilpotent Lie group
$\pi_{\omega}:\mathsf{G}\to U(\mathpzc{F}(V,\omega))$ by identifying $\omega$
with a central character, using that $\Gamma=\mathfrak{so}(V)\cap GL(V)$, and
by defining

$\pi_{\omega}(X)f(x):=\mathrm{e}^{-\frac{g_{\omega}(X,X)}{2}-\langle
x,X\rangle_{\omega}}f(x+X),$ (21)

for $X\in X$ and $f\in\mathpzc{F}(V,\omega)$. By construction, $\pi_{\omega}$
is a unitary irreducible representation representing the orbit
$\omega\in\Gamma$ under the Kirillov correspondence (beware that we follow the
convention of identifying $\Lambda$ with the set of flat orbits
$\Gamma\subseteq\mathfrak{g}^{*}/\mathsf{Ad}^{*}$). Since this construction
depends continuously on $\omega$, some arguments with the Bargmann transform
shows that the family of Hilbert spaces
$(\mathpzc{F}(V,\omega))_{\omega\in\Gamma}$ and representations
$(\pi_{\omega})_{\omega\in\Gamma}$ glues together to a locally trivial bundle
of Hilbert spaces $\mathcal{H}_{\Gamma}\to\Gamma$ with a strongly continuous
representation of $\mathsf{G}$.

The group $O(V)$ (defined with respect to the fixed inner product) is a
maximal compact subgroup of $\mathrm{Aut}_{\rm gr}(\mathsf{G})$ and acts on
the flat orbits $\Gamma$ by $\alpha.\omega:=\alpha^{t}\omega\alpha$. The
homeomorphism $\Gamma=GL(V)\cap\mathfrak{so}(V)\cong
GL(V)/\mathrm{Sp}(V,\omega_{0})$ is clearly $O(V)$-equivariant for the left
action on $GL(V)/\mathrm{Sp}(V,\omega_{0})$. The $O(V)$-action on $\Gamma$
lifts to a strongly continuously action on the bundle $\mathcal{H}_{\Gamma}$,
by simply letting $\alpha\in O(V)$ act as the unitary

$\alpha:\mathpzc{F}(V,\omega)\to\mathpzc{F}(V,\alpha.\omega),\;\alpha
f:=\alpha^{*}f.$

This map is well defined due to the fact that
$J_{\alpha.\omega}=\alpha^{-1}J_{\omega}\alpha$ and
$g_{\alpha.\omega}=\alpha^{t}g_{\omega}\alpha$, so the action is well defined
as soon as $\alpha^{t}=\alpha^{-1}$, i.e. $\alpha$ orthogonal.

We summarize this argument in the following proposition.

######  Proposition 0.9.21.

We let $\mathsf{G}$ denote free step $2$ nilpotent Lie group on the even-
dimensional inner product space $V$ with space of flat orbits
$\Gamma=GL(V)\cap\mathfrak{so}(V)$ being all symplectic forms on $V$. The
$O(V)$-equivariant bundle of Bargmann-Fock spaces
$\mathcal{H}_{\Gamma}\to\Gamma$ satisfies that

$I_{\mathsf{G}}\cong C_{0}(\Gamma,\mathbb{K}(\mathcal{H}_{\Gamma})),$

as $O(V)$-equivariant $C_{0}(\Gamma)-C^{*}$-algebras via the family
$(\pi_{\omega})_{\omega\in\Gamma}$. In particular,

$\begin{cases}[\zeta|_{O(V)}]=0\in
H^{1}(O(V),\check{H}^{2}(GL(V)\cap\mathfrak{so}(V),\mathbb{Z}))=0,\;\mbox{
and}\\\ c_{O(V)}=0\in
H^{2}(O(V),C(GL(V)\cap\mathfrak{so}(V),U(1))).\end{cases}$

####  0.9.3. Nilpotent Lie groups with one-dimensional center

Let us consider another class of nilpotent Lie groups, namely those with a
one-dimensional center. Given any nilpotent Lie group, a quotient by a
codimension one subgroup of the center produces a nilpotent Lie group with
one-dimensional center, making the case of one-dimensional center interesting.
We shall also consider a construction found in [112] that realize a nilpotent
Lie group as a closed subgroup of a nilpotent Lie group with one-dimensional
center admitting flat orbits – this example holds potential for index
theoretical considerations.

Assume that $\mathsf{G}$ is a simply connected nilpotent Lie group with one-
dimensional center, and as above $\mathfrak{g}$ denotes its Lie algebra that
we equipp with a choice of Jordan-Hölder basis such that $X_{1}$ is central.
As in Remark 0.6.7, there is an anti-symmetric mapping
$\omega:(\mathfrak{g}/\mathfrak{z})\wedge(\mathfrak{g}/\mathfrak{z})\to\mathfrak{z}$.
The group $\mathsf{G}$ admits flat orbits if and only if $\omega$ is non-
degenerate, and Remark 0.6.7 shows that

$\Gamma=\mathfrak{z}^{*}\setminus\\{0\\}.$

If $\mathsf{G}$ is graded, the associated dilation on $\mathfrak{z}$ is scalar
and we can identify $\Gamma_{\partial}=\\{\xi\in\Gamma:\xi(X_{1})=\pm 1\\}$.

To simplify our discussion, we assume that $\mathsf{G}$ is graded. If
$\xi_{0}\in\Gamma$ and $\mathfrak{h}$ is a real algebraic polarization of
$\xi_{0}$, we can extend $\mathfrak{h}$ to a bundle of polarizations of
$\Gamma$ by setting

$\mathfrak{h}(\xi):=\delta_{t}\mathfrak{h},\quad\mbox{for $t>0$ such that
$\xi=\pm\delta_{t}(\xi_{0})$.}$

If $\mathfrak{h}=\mathfrak{h}_{V}(\xi_{0})$ and the Jordan-Hölder basis is
homogeneous for the grading, we have that
$\mathfrak{h}(\xi)=\mathfrak{h}_{V}(\xi)$ for all $\xi\in\Gamma$. In general,
it is non-trivial to describe $\mathfrak{h}_{V}(\xi)$ beyond Proposition
0.6.8. But we note the following concerning the ideal $I\subseteq
C^{*}(\mathsf{G})$, if $\mathfrak{h}$ is any real algebraic polarization at
$\xi_{0}\in\Gamma$, and $\pi_{\pm}:\mathsf{G}\to U(L^{2}(\mathbb{R}^{d}))$
denotes the representations associated with $\pm\xi_{0}$ (polarized by
$\mathfrak{h}$) and $d=\mathrm{codim}(\mathfrak{z})/2=(\dim(\mathsf{G})-1)/2$,
the dilation induces an explicit isomorphism

$I_{\mathsf{G}}\xrightarrow{\sim}C_{0}(\mathfrak{z}^{*}\setminus\\{0\\},\mathbb{K}(L^{2}(\mathbb{R}^{d}))),\quad
a\mapsto(\xi\mapsto\pi_{\mathrm{sign}(\xi)}(\delta_{t(\xi)}(a))),$

where $\mathrm{sign}(\xi)\in\\{\pm\\}$ and $t(\xi)>0$ are determined by
$\xi=\mathrm{sign}(\xi)t(\xi)\xi_{0}$. For nilpotent Lie groups with one-
dimensional center, the ideal of flat orbits fits into an
$\mathrm{Aut}(\mathfrak{g}$-equivariant short exact sequence

$0\to
C_{0}(\mathfrak{z}^{*}\setminus\\{0\\},\mathbb{K}(L^{2}(\mathbb{R}^{d})))\to
C^{*}(\mathsf{G})\to C^{*}(\mathsf{G}/\mathsf{Z})\to 0.$

######  Example 0.9.22.

We start with an example of a nilpotent Lie group with one-dimensional center
lacking flat orbits. Consider the $4$-dimensional Lie algebra $\mathfrak{g}$
spanned by $Y_{1}$, $Y_{2}$, $Y_{3}$, and $Y_{4}$ subject to the relations

$[Y_{2},Y_{4}]=Y_{1}\quad\mbox{and}\quad[Y_{3},Y_{4}]=Y_{2},$

and all other Lie brackets between the basis vectors are zero. This example
was studied in [40, Example 1.3.10 and 2.2.2] and [97, Chapter 3.3]. We grade
$\mathfrak{g}$ by
$\mathfrak{g}=\mathfrak{g}_{-3}\oplus\mathfrak{g}_{-2}\oplus\mathfrak{g}_{-1}$
where $\mathfrak{g}_{-1}$ is spanned by $Y_{3},Y_{4}$, $\mathfrak{g}_{-2}$ is
spanned by $Y_{2}$ and the center $\mathfrak{g}_{-3}=\mathfrak{z}$ is spanned
by $Y_{1}$. The Lie algebra $\mathfrak{g}$ is step $3$ nilpotent with one-
dimensional center. The associated Lie group $\mathsf{G}$ readily seen to be a
subgroup of $\mathsf{G}_{3}$ (see Example 0.9.23). A short computation shows
that $\mathrm{Aut}_{\rm
gr}(\mathfrak{g})\cong\mathbb{R}\rtimes(\mathbb{R}^{\times})^{2}$ with action
induced from an action on $\mathfrak{g}_{-1}$, so the Jordan-Hölder flag
defined from $Y_{1},Y_{2},Y_{3},Y_{4}$ is invariant under graded
automorphisms, so computations in this basis globalizes in a controlled way.
Since the centre has odd codimension there exists no flat orbits. The top
coarse strata will however consist of generic orbits and produce structures
similar to that of the flat orbits. We include this example to indicate how
the techniques of this monograph can be extended beyond the usage of flat
orbits.

We use $Y_{1},Y_{2},Y_{3},Y_{4}$ as a Jordan-Hölder basis and we use the dual
basis for $\mathfrak{g}^{*}$ with associated coordinates
$\xi_{1},\xi_{2},\xi_{3},\xi_{4}$. Following [40, Example 1.3.10 and 2.2.2]
and [97, Chapter 3.3], the embedding $(p,q)\mapsto(0,\frac{q}{2p},0,p)$
identify

$\tilde{\Gamma}:=\mathbb{R}^{\times}_{p}\times\mathbb{R}_{q},$

with an open, dense Hausdorff subset of $\hat{\mathsf{G}}$ lifting to the
subset $\\{\xi:\xi_{4}\neq 0\\}$. under the projection map
$\mathfrak{g}^{*}\to\hat{\mathsf{G}}$. The inverse to the identification is
given by $p(\xi_{1},\xi_{2},\xi_{3},\xi_{4})=\xi_{1}$ and
$q(\xi_{1},\xi_{2},\xi_{3},\xi_{4})=2\xi_{3}\xi_{1}-\xi_{2}^{2}$. Each element
$\xi\in\tilde{\Gamma}$ defines a two-dimensional coadjoint orbit and is
polarized by the abelian subalgebra
$\mathfrak{h}=\mathbb{R}Y_{1}+\mathbb{R}Y_{2}+\mathbb{R}Y_{3}$. For
dimensional reasons, $\tilde{\Gamma}$ can be identified with the top coarse
strata in $\hat{G}$ and $\tilde{\Gamma}$ is preserved by all of
$\mathrm{Aut}(\mathsf{G})$. For $\xi\in\tilde{\Gamma}$, we construct the
representation

$\pi_{\xi}:\mathsf{G}\to
U(L^{2}(\mathbb{R})),\quad\pi_{\xi}:=\mathrm{i}\mathrm{n}\mathrm{d}\,_{\mathsf{H}}^{\mathsf{G}}\chi_{\xi},$

where $\mathsf{H}$ is the Lie group constructed from $\mathfrak{h}$,
$\chi_{\xi}:H\to U(1)$ the character defined from $\xi$ and where we have
identified $\mathsf{G}/\mathsf{H}=\mathbb{R}$.

Let $\tilde{I}_{\mathsf{G}}\subseteq C^{*}(\mathsf{G})$ denote the ideal
corresponding to $\tilde{\Gamma}$. The arguments above shows that the family
$(\pi_{\xi})_{\xi\in\tilde{\Gamma}}$ induces an
$\mathrm{Aut}(\mathsf{G})$-equivariant isomorphism

$\tilde{I}_{\mathsf{G}}\cong
C_{0}(\tilde{\Gamma},\mathbb{K}(L^{2}(\mathbb{R}))),$

for some projective $\mathrm{Aut}(\mathsf{G})$-action on $L^{2}(\mathbb{R})$.
In particular, $\tilde{I}_{\mathsf{G}}$ is an
$\mathrm{Aut}(\mathsf{G})$-continuous trace algebra with $\delta_{\rm
DD}(\tilde{I}_{\mathsf{G}})=0\in\check{H}^{3}(\tilde{\Gamma},\mathbb{Z})$ and
$[\zeta]=0\in
H^{1}(\mathrm{Aut}(\mathsf{G}),\check{H}^{2}(\tilde{\Gamma},\mathbb{Z}))$.

######  Example 0.9.23.

We proceed with a family of nilpotent Lie groups with one-dimensional center
lacking flat orbits. Let $\mathsf{G}_{n}$ denote the nilpotent Lie group of
real $(n+1)\times(n+1)$ upper triangular unipotent matrices. This is a
nilpotent Lie group of step length $n$. Its center is one-dimensional and
consists of matrices $g=(g_{ij})_{i,j=1}^{n+1}$ such that $g_{ij}=0$ unless
$i=j$ or $i=1$ and $j=n+1$ (the rigthmost, upmost entry can be non-zero). For
any $\xi\in\mathfrak{z}^{*}\setminus\\{0\\}$, the Lie algebra
$\mathsf{stab}(\xi)$ consists of matrices with zero entries on and below the
diagonal as well as in the right most column with the exception of the
rigthmost, upmost entry. In particular,
$\mathrm{codim}(\mathsf{stab}(\xi))=n-1<\mathrm{codim}(\mathfrak{z})$, so
there exists no flat orbits on $\mathsf{G}_{n}$ if $n>2$ even if
$\mathrm{codim}(\mathfrak{z})$ for certain $n$ is even-dimensional.

The case $n=3$ was computed in more detail in [40, Example 1.3.11 and 2.2.3].
In the same way as in Example 0.9.22, [40, Example 1.3.11 and 2.2.3] provides
an open dense Hausdorff subset
$\tilde{\Gamma}\cong\mathbb{R}^{\times}\times\mathbb{R}$ of
$\hat{\mathsf{G}}_{3}$. Each coadjoint orbit defined from elements of
$\tilde{\Gamma}$ are of dimension $4$ and the points of $\tilde{\Gamma}$ are
all polarized by the same $4$-dimensional abelian ideal $\mathfrak{h}$. For
dimensional reasons, $\tilde{\Gamma}$ can be identified with the top coarse
strata in $\hat{G}_{3}$ and $\tilde{\Gamma}$ is preserved by all of
$\mathrm{Aut}(\mathsf{G}_{3})$. Let $\tilde{I}_{\mathsf{G}_{3}}\subseteq
C^{*}(\mathsf{G}_{3})$ denote the ideal corresponding to $\tilde{\Gamma}$. The
arguments above shows that the family $(\pi_{\xi})_{\xi\in\tilde{\Gamma}}$
induces an $\mathrm{Aut}(\mathsf{G}_{3})$-equivariant isomorphism

$\tilde{I}_{\mathsf{G}_{3}}\cong
C_{0}(\tilde{\Gamma},\mathbb{K}(L^{2}(\mathbb{R}^{2}))),$

for some projective $\mathrm{Aut}(\mathsf{G}_{3})$-action on
$L^{2}(\mathbb{R}^{2})$. In particular, $\tilde{I}_{\mathsf{G}}$ is an
$\mathrm{Aut}(\mathsf{G}_{3})$-continuous trace algebra with $\delta_{\rm
DD}(\tilde{I}_{\mathsf{G}_{3}})=0\in\check{H}^{3}(\tilde{\Gamma},\mathbb{Z})$
and $[\zeta]=0$ in the group
$H^{1}(\mathrm{Aut}(\mathsf{G}_{3}),\check{H}^{2}(\tilde{\Gamma},\mathbb{Z}))$.
It would be interesting to extend this construction to $n>3$ as it relates to
parabolic geometries for $SL(n,\mathbb{R})$ (see Example 0.16.11 below).

######  Example 0.9.24 (The Mohsen construction).

We will now consider a construction from [112] that to each simply connected
nilpotent Lie group associates a larger simply connected Lie group with one-
dimensional center and flat orbits. This construction is of interest since it
creates flat orbits, and the modification is functorial enough to extend to
bundle of Lie groups and to Carnot manifolds.

Consider a simply connected nilpotent Lie group $\mathsf{G}$ with Lie algebra
$\mathfrak{g}$. The Lie algebra acts via the coadjoint action as Lie algebra
homomorphisms on $\mathfrak{g}^{*}$ equipped with the trivial Lie bracket. We
can form the Lie algebra
$\mathfrak{g}^{*}\rtimes_{\mathsf{ad}^{*}}\mathfrak{g}$. More precisely, for
$(\eta,X),(\eta^{\prime},X^{\prime})\in\mathfrak{g}^{*}\rtimes_{\mathsf{ad}^{*}}\mathfrak{g}:=\mathfrak{g}^{*}\oplus\mathfrak{g}$
their Lie bracket is defined by

$[(\eta,X),(\eta^{\prime},X^{\prime})]_{\mathfrak{g}^{*}\rtimes_{\mathsf{ad}^{*}}\mathfrak{g}}:=(\mathsf{ad}^{*}(X)\eta^{\prime}-\mathsf{ad}^{*}(X^{\prime})\eta,[X,X^{\prime}]).$

The duality pairing defines a symplectic form $\omega_{\mathfrak{g}}$ on
$\mathfrak{g}^{*}\oplus\mathfrak{g}$ which is readily verified to defined a
cocycle on $\mathfrak{g}^{*}\rtimes_{\mathsf{ad}^{*}}\mathfrak{g}$. We let
$\tilde{\mathfrak{g}}$ denote the associated central extension. More
precisely,
$\tilde{\mathfrak{g}}:=\mathfrak{g}^{*}\oplus\mathfrak{g}\oplus\mathbb{R}$ as
a vector space and the Lie bracket of
$((\eta,X),t),((\eta^{\prime},X^{\prime}),t^{\prime})\in\tilde{\mathfrak{g}}=\mathfrak{g}^{*}\oplus\mathfrak{g}\oplus\mathbb{R}$
is defined by

$[((\eta,X),t),((\eta^{\prime},X^{\prime}),t^{\prime})]_{\tilde{\mathfrak{g}}}:=([(\eta,X),(\eta^{\prime},X^{\prime})]_{\mathfrak{g}^{*}\rtimes_{\mathsf{ad}^{*}}\mathfrak{g}},\omega_{\mathfrak{g}}((\eta,X),(\eta^{\prime},X^{\prime}))).$

Let $\tilde{\mathsf{G}}$ denote the simply connected Lie group constructed
from $\tilde{\mathfrak{g}}$. The step length of $\tilde{\mathsf{G}}$ is
exactly one step larger than that of $\mathsf{G}$.

######  Definition 0.9.25.

Let $\mathsf{G}$ be a simply connected nilpotent Lie group with Lie algebra
$\mathfrak{g}$. The Lie algebra $\tilde{\mathfrak{g}}$ constructed in the
paragraph above is called the Mohsen modification of $\mathfrak{g}$ and the
Lie group $\tilde{\mathsf{G}}$ is called the Mohsen modification of
$\mathsf{G}$.

It is readily verified that the center $\tilde{\mathfrak{z}}$ of
$\tilde{\mathfrak{g}}$ is the one-dimensional summand
$\mathbb{R}\subseteq\tilde{\mathfrak{g}}$ and that the antisymmetric mapping
$\omega:\tilde{\mathfrak{g}}/\tilde{\mathfrak{z}}\wedge\tilde{\mathfrak{g}}/\tilde{\mathfrak{z}}\to\tilde{\mathfrak{z}}$
can be identified with the symplectic pairing $\omega_{\mathfrak{g}}$. In
particular, we can from the discussion in the start of this subsubsection
deduce that $\tilde{\mathfrak{g}}$ is a Lie algebra with one-dimensional
center that admits flat orbits

$\Gamma=\mathbb{R}^{\times}=\tilde{\mathfrak{z}}^{*}\setminus\\{0\\}.$

The automorphism group $\mathrm{Aut}(\tilde{\mathfrak{g}})$ is slightly
convoluted to describe and for our purposes it suffices to note that there are
injections

$\mathrm{Aut}(\mathfrak{g})\hookrightarrow\mathrm{Aut}(\mathfrak{g}^{*}\rtimes_{\mathsf{ad}^{*}}\mathfrak{g})\quad\mbox{and}\quad\mathrm{Aut}(\mathfrak{g})\hookrightarrow\mathrm{Aut}(\tilde{\mathfrak{g}}).$
(22)

The Lie algebra $\tilde{\mathfrak{g}}$ can always be graded by
$\tilde{\mathfrak{g}}_{-1}:=\mathfrak{g}^{*}\oplus\mathfrak{g}$ and
$\tilde{\mathfrak{g}}_{-2}:=\mathbb{R}$. We call this grading the standard
grading on $\tilde{\mathfrak{g}}$. In examples, when $\mathfrak{g}$ is graded
it is more natural to consider the induced grading on $\tilde{\mathfrak{g}}$.
Let us describe how to induce gradings from $\mathfrak{g}$ to
$\tilde{\mathfrak{g}}$ (this construction is also found in [112, Subsection
3.1]). If $\mathfrak{g}=\bigoplus_{-n\leq p<0}\mathfrak{g}_{p}$ is a grading
of $\mathfrak{g}$, we define

$\tilde{\mathfrak{g}}_{p}:=\begin{cases}0,&p<-n-1,\\\ \mathbb{R},&p=-n-1,\\\
\mathfrak{g}_{p}\oplus\mathfrak{g}^{*}_{-n-p-1},&p\geq-n,\end{cases}$

In the standard grading
$\mathrm{Aut}(\mathfrak{g})\hookrightarrow\mathrm{Aut}_{\rm
gr}(\tilde{\mathfrak{g}})$ and in an induced grading $\mathrm{Aut}_{\rm
gr}(\mathfrak{g})\hookrightarrow\mathrm{Aut}_{\rm gr}(\tilde{\mathfrak{g}})$.

We polarize $\xi_{0}=1\in\mathbb{R}^{*}\setminus\\{0\\}$ by the real algebraic
polarization

$\mathfrak{h}:=\mathbb{R}\oplus\mathfrak{g}^{*}.$

For a suitable Jordan-Hölder basis and either the standard grading or an
induced grading on $\tilde{\mathfrak{g}}$, the dilation reconstructs the
Vergne polarization of the flat orbits from this polarization and
$\mathfrak{h}_{V}(\xi)=\mathbb{R}\oplus\mathfrak{g}^{*}$ for any
$\xi\in\Gamma$. Let us describe the flat representations constructed from the
polarization $\mathfrak{h}:=\mathbb{R}\oplus\mathfrak{g}^{*}$. We have that

$\mathcal{H}_{V}=\mathbb{R}^{\times}\times L^{2}(\mathsf{G}),$

and the representation in the fibre over $\xi\in\mathbb{R}^{\times}$ is given
by

$\displaystyle\pi_{\xi}$
$\displaystyle\left(\mathrm{e}^{\eta+z}g\right)f(t):=\mathrm{e}^{i\xi\left(z+\eta(\log(t))\right)}f(gt),$

for $g\in\mathsf{G}$, $\eta\in\mathfrak{g}^{*}$,
$z\in\mathbb{R}=\tilde{\mathfrak{z}}$ and $f\in L^{2}(\mathsf{G})$. The reader
should note that the restriction of any flat orbit representation of
$\tilde{\mathsf{G}}$ to $\mathsf{G}$ is equivalent to the left regular
representation of $\mathsf{G}$.

We arrive at an isomorphism $I_{\tilde{\mathsf{G}}}\cong
C_{0}(\mathbb{R}^{\times},\mathbb{K}(L^{2}(\mathsf{G})))$ and a short exact
sequence

$0\to C_{0}(\mathbb{R}^{\times},\mathbb{K}(L^{2}(\mathsf{G})))\to
C^{*}(\tilde{\mathsf{G}})\to
C^{*}(\mathfrak{g}^{*}\rtimes_{\mathsf{Ad}^{*}}\mathsf{G})\to 0.$

The action of $\mathrm{Aut}(\tilde{\mathsf{G}})$ on $I_{\tilde{\mathsf{G}}}$
restricts to the ordinary fibrewise $\mathrm{Aut}(\mathsf{G})$-action on
$I_{\tilde{\mathsf{G}}}\cong
C_{0}(\mathbb{R}^{\times},\mathbb{K}(L^{2}(\mathsf{G})))$ under the mapping
$\mathrm{Aut}(\mathsf{G})\hookrightarrow\mathrm{Aut}(\tilde{\mathsf{G}})$ from
Equation (22). Furthermore, we deduce that $[\zeta]=0\in
H^{1}(\mathrm{Aut}(\tilde{\mathsf{G}}),\check{H}^{2}(\mathbb{R}^{\times},\mathbb{Z}))=0$
and $[c_{\mathrm{Aut}(\mathsf{G})}]=0\in
H^{2}(\mathrm{Aut}(\mathsf{G}),C(\mathbb{R}^{\times},U(1)))=0$.

##  Part II Groupoids and twists

Introduction to part 2

The aim of the current part is to extend the structures defined in the
previous part to bundles of nilpotent Lie groups and to prove the
$K$-theoretical results needed for further index theoretical considerations.
The motivation to do so comes from the fact that the tangent bundle of a
Carnot manifold can be seen as a groupoid globally describing the osculating,
nilpotent Lie group structure on each fibre. In the following table we gather
the main characters of the story and compare notation for their incarnations
on the level of the fiber and the level of frame bundle $P\to X$ of the bundle
of Lie groups:

Lie group (group theoretic model fiber) | $\mathsf{G}$ |  Locally trivial bundle of Lie groups (e.g. the tangent groupoid of a Carnot manifold) | $\mathcal{G}$
---|---|---|---
Lie algebra (Lie algebraic model fiber) | $\mathfrak{g}$ |  Lie algebroid with trivial anchor map (e.g. the tangent algebroid of a Carnot manifold) | $\mathbf{g}$
Center | $\mathsf{Z}$ |  Central groupoid | $\mathcal{Z}$
Set of flat orbits | $\Gamma$ |  Fiber bundle of flat orbits | $\Gamma_{P}$
Total space of flat orbits | $\Xi$ |  Symplectic vector bundle of flat orbits over the total space of the fiber bundle of flat orbits | $\Xi_{P}$
Ideal of flat representations | $I_{\mathsf{G}}$ |  Ideal of flat orbit representations in $C^{*}(\mathcal{G})$ | $I_{P}$

Later, when $P$ is constructed from a regular Carnot manifold $X$ we indicate
$X$ rather than $P$ in the notation.

Homological information about $H$-elliptic operators on Carnot manifolds is
naturally contained in $K_{*}(C^{*}(\mathcal{G}))$. The aim is to develop
computation tools in these $K$-groups. In particular, there will be three main
results in this part:

  1. (1)

The Connes-Thom isomorphism allow us to establish isomorphism
$K_{*}(C^{*}(\mathsf{G}))\simeq K_{*}(C_{0}(\mathfrak{g}^{*}))$, this is known
since work of Nistor [119]. This result extends to the level of bundles, i.e.
for a locally trivial bundle of nilpotent connected simply connected Lie
groups we show that $K_{*}(C^{*}(\mathcal{G}))\simeq
K_{*}(C_{0}(\mathbf{g}^{*}))$. The technique used to construct Nistor’s
Connes-Thom isomorphism is that of adiabatic deformations: a groupoid that
smoothly deforms $\mathcal{G}$ to $\mathbf{g}$.

  2. (2)

We prove that the procedure of Nistor above restricts from
$C^{*}(\mathcal{G})$ to $I_{P}$ and gives an isomorphism $K_{*}(I_{P})\simeq
K^{*}(\Xi_{P})$. Both isomorphisms for $C^{*}(\mathcal{G})$ and for $I_{P}$
are of an abstract nature and do not provide concrete recipe for transforming
cycles in $K_{*}(I_{P})$ to cycles in $K^{*}(\Xi_{P})$. Extending the proof
that $\delta(I_{\mathsf{G}})=0$ – by constructing a concrete Morita
equivalence bundle – to the case of a bundle of Lie groups we construct an
isomorphism
$K_{*}(I_{P})\xrightarrow{Mor}K^{*}(\Gamma_{P})\xrightarrow{Thom}K^{*}(\Xi_{P})$
from Morita invariance and the Thom isomorphism. This isomorphism coincides
with the Connes-Thom isomorphisms up to a line bundle called the metaplectic
correction bundle.

  3. (3)

Finally, we show that the map $K_{*}(I_{P})\rightarrow
K_{*}(C^{*}(\mathcal{G}))$, induced from inclusion, is surjective. Thus, the
classes of $H$-elliptic operators in $K_{*}(C^{*}(\mathcal{G}))$ can be
parametrized by $K^{*}(\Gamma_{P})$ as it is possible to lift to
$K_{*}(I_{P})$ and then use the Morita equivalence to transform to cycles in
$K^{*}(\Gamma_{P})$.

In Section 0.10 we review definitions, basic facts and examples from the
theory of Lie groupoids and algebroids. We recall the construction of the
$C^{*}$-algebra of a Lie groupoid and the definition of the Lie algebroid
associated with a Lie groupoid. In Section 0.11 we describe central
decomposition of a Lie group as a twisted semidirect product. This allows to
establish equivariant isomorphisms between $C^{*}(\mathsf{G})$ and
$C^{*}(\mathcal{G}_{\mathsf{Z}},\omega_{\mathsf{Z}})$, where
$\mathcal{G}_{\mathsf{Z}}$ is a certain groupoid over the dual of the center
$\mathfrak{z}^{*}$. For nilpotent groups this isomorphism restricts to the
ideal of flat orbits $I_{\mathsf{G}}$ and provides isomorphism
$I_{\mathsf{G}}\simeq C^{*}(\mathcal{G}_{\Gamma},\omega_{\Gamma})$ where
$\mathcal{G}_{\Gamma}=\mathcal{G}_{\mathsf{Z}}|_{\Gamma}$.

In Section 0.12 we extend the structures of Section 0.10 and 0.11 from
nilpotent Lie groups to bundles of nilpotent Lie groups. For instance, we
prove analogues of central decompositions considered in Section 0.11. In
Section 0.13 we construct Connes-Thom isomorphisms
$K_{*}(C^{*}(\mathcal{G}))\simeq K_{*}(C_{0}(\mathbf{g}^{*}))$ and
$K_{*}(I_{P})\simeq K^{*}(\Gamma_{P})$. The construction requires notions of
classical adiabatic groupoids, parabolically blowed adiabatic groupoid and a
cocycle deformation of the adiabatic groupoid. At the end of the section we
prove surjectivity of the map $K_{*}(I_{P})\rightarrow
K_{*}(C^{*}(\mathcal{G}))$ induced from the inclusion.

The results of Section 0.14 are those of most relevance for later results on
index theory. A crucial result in this section states that the Connes-Thom
isomorphism $K_{*}(I_{X})\simeq K^{*}(\Gamma_{X})$ differs from the Morita
equivalence defined from a bundle of flat orbit representations up to a line
bundle – the metaplectic correction bundle. This result lies at the heart of
the index theorem as it allow us to explicitly compute the Connes-Thom map
$K_{*}(C^{*}(\mathsf{G}))\to K_{*}(C_{0}(\mathfrak{g}^{*}))$ by finding a
preimage under $K_{*}(I_{P})\rightarrow K_{*}(C^{*}(\mathcal{G}))$ and the
applying Morita invariance.

###  0.10. Lie groupoids and their $C^{*}$-algebras

A groupoid can abstractly be defined as a small category in which all
morphisms are isomorphisms. The abstract definition says quite litte and
should rather be interpreted as having a space (the objects) with symmetries
(the morphisms) that could vary over the space. We shall solely be concerned
with Lie groupoids and twists of them.

We will recall the basic definitions for groupoids for the purpose of setting
notations. The reader is expected to be acquainted with groupoids. The
references [48, 49, 51, 111, 120, 149, 150] can be consulted for a more
extensive overview of groupoids.

######  Definition 0.10.1.

Let $M$ be a smooth manifold. A Lie groupoid $\mathcal{G}\rightrightarrows M$
over $M$ is a smooth manifold of morphisms $\mathcal{G}$ equipped with the
following structure maps:

  1. (1)

A source and range maps $s,r:\mathcal{G}\rightarrow M$.

  2. (2)

A composition map $\circ:\mathcal{G}^{(2)}\rightarrow\mathcal{G}$, where

$\mathcal{G}^{(2)}=\\{(\gamma_{1},\gamma_{2})\in\mathcal{G}\times\mathcal{G}:r(\gamma_{2})=s(\gamma_{1})\\},$

which is associative in the sense that
$(\gamma_{1}\circ\gamma_{2})\circ\gamma_{3}=\gamma_{1}\circ(\gamma_{2}\circ\gamma_{3})$
whenever all compositions are defined.

  3. (3)

A unit map $e:M\rightarrow\mathcal{G}$, subject to $re=se=e$, that maps each
object to the identity morphism in the sense that
$e(r(\gamma))\circ\gamma=\gamma\circ e(s(\gamma))=\gamma$ for each $\gamma\in
G$.

  4. (4)

An inverse map $i:\mathcal{G}\rightarrow\mathcal{G}$, subject to $ri=s$,
$si=r$ and $i^{2}=\mathrm{id}_{\mathcal{G}}$, which sends each morphism to its
inverse in the sense that $i(\gamma)\circ\gamma=e(s(\gamma))$ and $\gamma\circ
i(\gamma)=e(r(\gamma))$.

All structure maps are required to be smooth and $s,r$ are required to be
submersions.

A Lie group is a Lie groupoid over a point. In the same way that a Lie group
has a Lie algebra, a Lie groupoid will have a Lie algebroid.

######  Definition 0.10.2.

A Lie algebroid over a smooth manifold $M$ is a triple
$(\mathbf{g},\rho,[\cdot,\cdot])$ where

  * •

$\mathbf{g}\rightarrow M$ is a vector bundle.

  * •

$\rho:\mathbf{g}\rightarrow TM$ is a vector bundle map called the anchor map.

  * •

$[\cdot,\cdot]:C^{\infty}(M,\mathbf{g})\times
C^{\infty}(M,\mathbf{g})\rightarrow C^{\infty}(M,\mathbf{g})$ is a Lie bracket
such that

    1. (1)

The map on sections induced from the anchor map
$\rho:C^{\infty}(M,\mathbf{g})\rightarrow C^{\infty}(M,TM)$ is a Lie algebra
homomorphism.

    2. (2)

For any $X,Y\in C^{\infty}(M,\mathbf{g})$, $f\in C^{\infty}(M)$ one has

$[X,fY]=f[X,Y]+(\rho(X)f)Y.$

######  Definition 0.10.3.

Let $\mathcal{G}\rightrightarrows M$ be a Lie groupoid over $M$. The Lie
algebroid $\mathsf{Lie}(\mathcal{G})$ of $\mathcal{G}$ is defined as follows:

  1. (1)

The vector bundle $\mathsf{Lie}(\mathcal{G})=\ker Ds|_{M}$ with Lie bracket
defined from identifying sections of $\mathsf{Lie}(\mathcal{G})$ with right
invariant vector fields on $\mathcal{G}$.

  2. (2)

The anchor map $\rho=Dr:\mathsf{Lie}(\mathcal{G})\rightarrow TM$.

######  Remark 0.10.4.

Under certain conditions one has an analogue of Lie’s Third Theorem, which
associates a Lie groupoid $\mathcal{G}$ with a Lie algebroid $\mathbf{g}$ such
that $\mathsf{Lie}(\mathcal{G})=\mathbf{g}$. In general, it can fail, see the
discussion in [48, Chapter 5.6].

Let us describe some examples that will play an important role later on in the
monograph.

######  Example 0.10.5 (Pair groupoid).

Let $M$ be a smooth manifold. The manifold $\mathcal{G}:=M\times M$ is a
groupoid over $M$ by setting

$\displaystyle r(x,y)$ $\displaystyle:=x,\quad s(x,y):=y,$
$\displaystyle(x,y)\circ(y,z)$ $\displaystyle:=(x,z),$ $\displaystyle i(x,y)$
$\displaystyle:=(y,x),\quad e(x):=(x,x).$

######  Example 0.10.6 (Transformation groupoid).

Given a Lie group $G$ acting by diffeomorphisms on a smooth manifold $M$,
define the Lie groupoid $M\rtimes G:=M\times G\rightrightarrows M$ with
operations

$\displaystyle r(x,g)$ $\displaystyle:=x,\quad s(x,g):=gx,$
$\displaystyle(x,g)\circ(xg,h)$ $\displaystyle:=(x,gh),$ $\displaystyle
i(x,g)$ $\displaystyle:=(xg,g^{-1}),\quad e(x):=(x,e_{G}).$

######  Example 0.10.7 (Deformation groupoid).

For a closed smooth submanifold $M_{0}$ of a smooth manifold $M$ with a normal
bundle $N$. The deformation to the normal cone is the smooth manifold

$D(M_{0},M)=N\times\\{0\\}\sqcup M\times(0,1].$

See [50] for the definition of the smooth structure on $D(M_{0},M)$. A smooth
groupoid $\mathcal{G}$ is called a deformation groupoid if
$\mathcal{G}=D(\mathcal{H}_{0},\mathcal{H})$ for a groupoid $\mathcal{H}$ and
closed Lie subgroupoid $\mathcal{H}_{0}$ with object space $M$. $\mathcal{G}$
has object space $M\times[0,1]$. For more details on this construction and the
next, see [50]. We occasionally consider the deformation groupoid construction
over $[0,\infty)$ instead of $[0,1]$.

######  Example 0.10.8 (Adiabatic groupoid).

Given a smooth groupoid $\mathcal{G}\rightrightarrows M$ one can consider its
Lie algebroid $\mathsf{Lie}(\mathcal{G})$ (see definition 0.10.3).
$\mathsf{Lie}(\mathcal{G})$ is considered to be a Lie groupoid with the vector
bundle operation. It is isomorphic to the normal bundle of unit the inclusion
$M\hookrightarrow\mathcal{G}$. The adiabatic groupoid $\mathcal{G}_{ad}$ is
defined to be

$\mathcal{G}_{\rm
ad}=D(M,\mathcal{G})\equiv\mathsf{Lie}(\mathcal{G})\times\\{0\\}\sqcup\mathcal{G}\times(0,1].$

######  Example 0.10.9 (Tangent groupoid).

The tangent groupoid $\mathbb{T}M\rightrightarrows M\times[0,1]$ of a smooth
manifold $M$ is defined to be adiabatic groupoid of the pair groupoid $M\times
M\rightrightarrows M$. It was first studied by Connes [39] as a way of
conceptualizing the Atiyah-Singer index theorem. The Lie algebroid of
$\mathbb{T}M$ is $TM\times[0,1]$. An important variation of the tangent
groupoid is the parabolic tangent groupoid of a Carnot manifold constructed by
van Erp-Yuncken (see [149]), we review it below in the section on Carnot
manifolds, see Section 0.17.

####  0.10.1. Twisted groupoid $C^{*}$-algebras

To encode the representations of a groupoid, one associates a $C^{*}$-algebra.
In order to be able to describe the full range of situations arising for
nilpotent groups and Carnot manifolds, we will need the generality of twisted
groupoid $C^{*}$-algebras. Before defining these $C^{*}$-algebras, we recall
some basic notions for groupoids.

Given a smooth groupoid $\mathcal{G}\rightrightarrows M$, for $x\in M$ we
denote the $s$-fiber of $\mathcal{G}$ by

$\mathcal{G}_{x}:=s^{-1}(\gamma)=\\{\gamma\in\mathcal{G}:s(\gamma)=x\\}.$

By definition, $\mathcal{G}_{x}$ is a smooth submanifold of $\mathcal{G}$.
Similarly, $\mathcal{G}^{x}:=r^{-1}(x)$ denotes the $r$-fibre in $x\in M$. For
any $\eta\in\mathcal{G}_{x}\cap\mathcal{G}^{y}$ there is an associated
diffeomorphism

$R_{\eta}:\mathcal{G}_{y}\rightarrow\mathcal{G}_{x},\
R_{\eta}(\gamma)=\gamma\circ\eta.$

######  Definition 0.10.10.

Let $\mathcal{G}\rightrightarrows M$ be a Lie groupoid. A Haar system on
$\mathcal{G}$ is a family of smooth measures $(\mu_{x})_{x\in M}$, where
$\mu_{x}$ is a measure on the $s$-fiber $\mathcal{G}_{x}$, such that

  1. (1)

For any $f\in C_{c}^{\infty}(\mathcal{G})$, the function $M\ni
x\mapsto\int_{\mathcal{G}_{x}}fd\mu_{x}\in\mathbb{C}$ is smooth on $M$.

  2. (2)

For any $\eta\in\mathcal{G}_{x}\cap\mathcal{G}^{y}$, the diffeomorphism
$R_{\gamma}:G_{y}\rightarrow G_{x}$ is measure-preserving.

We note that the second condition in the definition of a Haar system encodes
that our convention is to use right invariant Haar system. Any two Haar
systems on a Lie groupoid $\mathcal{G}\rightrightarrows M$ differ by a smooth,
positive function on $M$. Therefore, the Haar system is essentially unique.

######  Definition 0.10.11.

A smooth 2-cocycle on a Lie groupoid $\mathcal{G}$ is a smooth map
$\omega:\mathcal{G}^{(2)}\rightarrow U(1)$ such that

$\omega(x,y)\omega(xy,z)=\omega(x,yz)\omega(y,z)\text{ whenever }(x,y),\
(y,z)\in\mathcal{G}^{(2)},$ $\omega(x,s(x))=1=\omega(r(x),x).$

######  Definition 0.10.12.

Let $\mathcal{G}\rightrightarrows M$ be a Lie groupoid and $\omega$ a smooth
$2$-cocycle on $\mathcal{G}$. The twisted convolution algebra of
$(\mathcal{G},\omega)$ is the $*$-algebra
$C_{c}^{\infty}(\mathcal{G},\omega):=C^{\infty}_{c}(\mathcal{G})$ of compactly
supported smooth functions on $\mathcal{G}$ equipped with the product defined
from twisted convolution

$f\ast
g(\gamma)=\int_{\mathcal{G}_{r(\gamma)}}\omega(\eta,\eta^{-1}\gamma)f(\eta)g(\eta^{-1}\gamma)\mathrm{d}\mu^{r(\gamma)},\quad
f,g\in C^{\infty}_{c}(\mathcal{G},\omega),$

and the involution is given by

$f^{*}(\gamma)=\overline{f(\gamma^{-1})\omega(\gamma,\gamma^{-1})}.$

######  Remark 0.10.13.

We say that a Lie groupoid $\mathcal{G}\rightrightarrows M$ is a polynomial
Lie groupoid if $M$ and $\mathcal{G}$ are manifolds with polynomial structures
(see [123]), and all maps involved in the groupoid structure are polynomial.
For a smooth $2$-cocycle $\omega$ on a polynomial Lie groupoid the Schwartz
space $\mathcal{S}(\mathcal{G},\omega):=\mathcal{S}(\mathcal{G})$ forms a
$*$-algebra with the identically defined operations as that of
$C_{c}^{\infty}(\mathcal{G},\omega)$. There is in this case an inclusion
$C_{c}^{\infty}(\mathcal{G},\omega)\subseteq\mathcal{S}(\mathcal{G},\omega)$
which defines a $*$-homomorphism with dense range.

Restriction along the unit defines a mapping
$E_{M}:C^{\infty}_{c}(\mathcal{G},\omega)\to C^{\infty}_{c}(M)$. For
$f_{1},f_{2}\in C^{\infty}_{c}(\mathcal{G},\omega)$ we define $\langle
f_{1},f_{2}\rangle_{C_{0}(M)}\in C^{\infty}_{c}(M)$ by

$\langle f_{1},f_{2}\rangle_{C_{0}(M)}:=E_{M}(f_{1}^{*}\ast f_{2}).$

Let $\mathpzc{E}_{\mathcal{G},\omega}$ denote the completion of
$C^{\infty}_{c}(\mathcal{G},\omega)$ as a $C_{0}(M)$-Hilbert $C^{*}$-module.
Up to natural isomorphism of $C_{0}(M)$-Hilbert $C^{*}$-modules,
$\mathpzc{E}_{\mathcal{G},\omega}$ is independent of $\omega$. Left
multiplication extends to a faithful $*$-representation

$\lambda^{\omega}:C^{\infty}_{c}(\mathcal{G},\omega)\to\mathrm{End}_{C_{0}(M)}^{*}(\mathpzc{E}_{\mathcal{G},\omega}).$

We call $\lambda$ the left regular representation of $\mathcal{G}$.

######  Definition 0.10.14.

Let $\mathcal{G}\rightrightarrows M$ be a Lie groupoid and $\omega$ a smooth
$2$-cocycle on $\mathcal{G}$. The reduced $C^{*}$-algebra
$C_{r}^{*}(\mathcal{G},\omega)$ is the completion of
$C_{c}^{\infty}(\mathcal{G},\omega)$ with respect to the norm

$\|f\|_{C_{r}^{*}(\mathcal{G},\omega)}:=\|\lambda^{\omega}(f)\|_{\mathrm{End}_{C_{0}(M)}^{*}(\mathpzc{E}_{\mathcal{G},\omega})}.$

The full $C^{*}$-algebra $C^{*}(\mathcal{G},\omega)$ is a completion of
$C_{c}^{\infty}(\mathcal{G},\omega)$ with respect to all $*$-representations
by bounded operators on a Hilbert space (for more details see [133]).

If $\omega\equiv 1$, we write $C_{r}^{*}(\mathcal{G})$ and
$C^{*}(\mathcal{G})$ instead of $C_{r}^{*}(\mathcal{G},1)$ and
$C^{*}(\mathcal{G},1)$, respectively.

######  Remark 0.10.15.

If $\mathcal{G}\rightrightarrows M$ is a polynomial Lie groupoid with a smooth
$2$-cocycle $\omega$, the inclusion
$C_{c}^{\infty}(\mathcal{G},\omega)\subseteq\mathcal{S}(\mathcal{G},\omega)$
extends to a continuous dense inclusion
$\mathcal{S}(\mathcal{G},\omega)\subseteq C^{*}_{r}(\mathcal{G},\omega)$.
Similarly, there is a continuous dense inclusion
$\mathcal{S}(\mathcal{G},\omega)\subseteq C^{*}(\mathcal{G},\omega)$.

######  Example 0.10.16.

Consider the transformation groupoid $\mathcal{G}=M\rtimes G$ associated with
a Lie group acting smoothly on a manifold. A choice of Haar system corresponds
to a choice of Haar measure on $G$. The convolution product on
$C_{c}^{\infty}(\mathcal{G})$ is given by

$f_{1}\ast f_{2}(x,g)=\int_{G}f_{1}(h,x)f_{2}(h^{-1}g,xh)\mathrm{d}h,$

where $\mathrm{d}h$ is the Haar measure on $G$. The involution is given by

$f^{*}(g,x)=\overline{f(g^{-1},xg)}.$

Therefore $C_{r}^{*}(\mathcal{G})=C_{0}(M)\rtimes_{r}G$ and
$C^{*}(\mathcal{G})=C_{0}(M)\rtimes G$.

######  Example 0.10.17.

Let $M$ be a smooth manifold and consider the pair groupoid
$\mathcal{G}=M\times M$. A Haar system on $\mathcal{G}$ corresponds to a
choice of a positive Radon measure on $M$. We suppress the choice of this
measure from the notations. The convolution on $C_{c}^{\infty}(\mathcal{G})$
is given by

$f_{1}\ast f_{2}(x,y)=\int_{M}f_{1}(x,z)f_{2}(z,y)\mathrm{d}z.$

The involution is given by

$f^{*}(x,y)=\overline{f(y,x)}.$

The reduced $C^{*}$-algebra of $\mathcal{G}$ is isomorphic to the
$C^{*}$-algebra of compact operators $\mathbb{K}(L^{2}(M))$. In fact, this
isomorphism is constructed by localizing the left regular representation in a
point $x\in M$ and the canonical left $\mathcal{G}$-equivariant diffeomorphism
$M\cong\mathcal{G}_{x}$.

######  Theorem 0.10.18.

Let $\mathcal{G}\rightrightarrows M$ be a Lie groupoid with a smooth
$2$-cocycle $\omega$. For any subset $X\subseteq M$ we define
$\mathcal{G}_{X}:=s^{-1}(X)\cap r^{-1}(X)$ which is a topological groupoid
(but not a Lie groupoid in general). If $X\subseteq M$ is open and
$\mathcal{G}$-invariant, $\mathcal{G}_{X}$ is an open Lie subgroupoid of
$\mathcal{G}$ and $C^{*}(\mathcal{G}_{X},\omega)$ is an ideal in
$C^{*}(\mathcal{G},\omega)$. If $X$ is a closed $\mathcal{G}$-invariant
submanifold, $\mathcal{G}_{X}$ is a closed Lie subgroupoid for which
restriction defines an epimorphism $C^{*}(\mathcal{G},\omega)\to
C^{*}(\mathcal{G}_{X},\omega)$ that fits into an exact sequence of full
groupoid $C^{*}$-algebras:

$0\rightarrow C^{*}(\mathcal{G}_{M\setminus X},\omega)\rightarrow
C^{*}(\mathcal{G},\omega)\rightarrow C^{*}(\mathcal{G}_{X},\omega)\rightarrow
0.$

We refer the proof of this theorem to [133]. The reader should beware that in
general, the exactness property in Theorem 0.10.18 does not hold for the
reduced completion. The following corollary is an important consequence that
we will use abundantly.

######  Corollary 0.10.19.

Let $\mathcal{G}\rightrightarrows M$ be a Lie groupoid. Assume that $\omega$
is a smooth $2$-cocycle on the adiabatic groupoid $\mathcal{G}_{ad}$. Let
$\mathcal{G}_{\rm ad}^{\circ}$ denote the restriction of $\mathcal{G}_{\rm
ad}$ to $M\times[0,1)$. Then one has the following exact sequence:

$0\rightarrow C^{*}(\mathcal{G}\times(0,1),\omega)\rightarrow
C^{*}(\mathcal{G}_{\rm ad}^{\circ},\omega)\xrightarrow{{\rm
ev}_{0}}C^{*}(\mathsf{Lie}(\mathcal{G}),\omega)\rightarrow 0.$

If $\omega=1$, the short exact sequence reduces to

$0\rightarrow C_{0}(0,1)\otimes C^{*}(\mathcal{G})\rightarrow
C^{*}(\mathcal{G}_{\rm ad}^{\circ})\xrightarrow{{\rm
ev}_{0}}C_{0}(\mathsf{Lie}(\mathcal{G})^{*})\rightarrow 0,$

where $\mathsf{Lie}(\mathcal{G})^{*}\to M$ denotes the dual vector bundle of
the Lie algebroid.

###  0.11. A groupoid description of the flat orbits

The ideal of flat orbits of a simply connected nilpotent Lie group can be
described as a twisted groupoid $C^{*}$-algebra. We shall see that this
relates the flat orbits to a noncommutative Fourier transform of functions on
the total space of the vector bundle $\Xi\to\Gamma$.

We start with some preliminary considerations. Consider a Lie group
$\mathsf{G}$ with a closed central subgroup $\mathsf{Z}\subseteq\mathsf{G}$.
For notational convenience, we use multiplicative notation on $\mathsf{G}$ and
$\mathsf{G}/\mathsf{Z}$ but additive notation on $\mathsf{Z}$. Assume that
$\tau:\mathsf{G}/\mathsf{Z}\to\mathsf{G}$ is a smooth splitting of the
quotient mapping $\mathsf{G}\to\mathsf{G}/\mathsf{Z}$. We define the smooth
$2$-cocycle

$\omega:\mathsf{G}/\mathsf{Z}\times\mathsf{G}/\mathsf{Z}\rightarrow\mathsf{Z},\
\omega(g_{1}\mathsf{Z},g_{2}\mathsf{Z}):=\tau(g_{1}\mathsf{Z})\tau(g_{2}\mathsf{Z})\tau(g_{1}g_{2}\mathsf{Z})^{-1}.$

######  Proposition 0.11.1.

The splitting $\tau$ induces an isomorphism
$\mathsf{G}\simeq\mathsf{G}/\mathsf{Z}\rtimes_{\omega}\mathsf{Z}$, where the
cocycle twisted product $\mathsf{G}/\mathsf{Z}\rtimes_{\omega}\mathsf{Z}$ is
the group defined as the set $\mathsf{G}/\mathsf{Z}\times\mathsf{Z}$ equipped
with the group operation

$(g_{1}\mathsf{Z},z_{1})\cdot_{\omega}(g_{2}\mathsf{Z},z_{2}):=(g_{1}g_{2}\mathsf{Z},z_{1}+z_{2}+\omega(g_{1}\mathsf{Z},g_{2}\mathsf{Z})).$

We are interested in the situation that $\mathsf{G}$ is a simply connected
nilpotent Lie group and $\mathsf{Z}$ is its center. We can define the
$2$-cocycle

$\omega_{Z}:\mathsf{G}/\mathsf{Z}\times\mathsf{G}/\mathsf{Z}\to\mathsf{Z},$
(23)

by choosing a Jordan-Hölder basis that splits
$\mathsf{G}\to\mathsf{G}/\mathsf{Z}$. The reader should note that this
$2$-cocycle pairs with $\mathfrak{z}^{*}$ to the Kirillov form on the flat
orbits, cf. Remark 0.6.7. By the discussion above we have that
$\mathsf{G}\cong\mathsf{G}/\mathsf{Z}\times_{\omega_{Z}}\mathsf{Z}$. Let us
translate this context into a twisted groupoid. The crucial observation is
that $\mathsf{Z}=\mathfrak{z}$ whose $C^{*}$-algebra is isomorphic to
$C_{0}(\mathfrak{z}^{*})$ via the Fourier transform.

######  Definition 0.11.2.

Let $\mathsf{G}$ be a simply connected nilpotent Lie group with center
$\mathsf{Z}=\mathfrak{z}$. Define the groupoid
$\mathcal{G}_{\mathsf{Z}}\rightrightarrows\mathfrak{z}^{*}$ as the
transformation groupoid (see Example 0.10.6) of the trivial
$\mathsf{G}/\mathsf{Z}$-action on $\mathfrak{z}^{*}$.

We write $\mathcal{G}_{\Gamma}$ for the restriction of
$\mathcal{G}_{\mathsf{Z}}$ to the open subset of flat orbits
$\Gamma\subseteq\mathfrak{z}^{*}$.

We note that
$\mathcal{G}_{\mathsf{Z}}^{(2)}=\mathsf{G}/\mathsf{Z}\times\mathsf{G}/\mathsf{Z}\times\mathfrak{z}^{*}$
and the composition is the product of the two $\mathsf{G}/\mathsf{Z}$-factors.

######  Definition 0.11.3.

Define the unitary 2-cocycle $\omega_{\mathcal{G}_{\mathsf{Z}}}$ on
$\mathcal{G}_{\mathsf{Z}}$ by

$\omega_{\mathcal{G}_{\mathsf{Z}}}:\mathcal{G}_{\mathsf{Z}}^{(2)}=\mathsf{G}/\mathsf{Z}\times\mathsf{G}/\mathsf{Z}\times\mathfrak{z}^{*}\rightarrow
U(1),\
\omega_{\mathcal{G}_{Z}}(g\mathsf{Z},h\mathsf{Z},\xi)=e^{-i\xi(\omega_{\mathsf{Z}}(g\mathsf{Z},h\mathsf{Z}))}.$

The group $\mathrm{Aut}(\mathsf{G})$ acts as smooth groupoid automorphisms on
$\mathcal{G}_{\mathsf{Z}}$ by

$\varphi(g\mathsf{Z},\xi):=(\varphi(g)\mathsf{Z},(\varphi^{-1})^{*}\xi).$

We are here implicitly using that a groupoid automorphism preserves the
center. Let us describe a lift of this action to an action on
$C^{*}(\mathcal{G}_{\mathsf{Z}},\omega_{\mathcal{G}_{\mathsf{Z}}})$.

######  Proposition 0.11.4.

Let $\mathsf{G}$ be a simply connected nilpotent Lie group with center
$\mathsf{Z}$ and fix a smooth splitting
$\tau:\mathsf{G}/\mathsf{Z}\to\mathsf{G}$. For
$\varphi\in\mathrm{Aut}(\mathsf{G})$, define
$\tau_{\varphi}:=\varphi^{-1}\circ\tau\circ\varphi$ and

$b_{\varphi}:=\tau_{\varphi}\cdot\tau^{-1}:\mathsf{G}/\mathsf{Z}\to\mathsf{Z}.$

It holds that the group $\mathrm{Aut}(\mathsf{G})$ acts strongly continuously