# Generic irreducibility of parabolic induction for real reductive groups

D. RENARD  Centre de mathématiques Laurent Schwartz, Ecole Polytechnique,
91128 Palaiseau Cedex<EMAIL_ADDRESS>
###### Abstract.

Let $G$ be a real reductive linear group in the Harish-Chandra class. Suppose
that $P$ is a parabolic subgroup of $G$ with Langlands decomposition $P=MAN$.
We prove that if $\pi$ is an irreducible representation of $M$, then for a
generic character $\chi=\chi_{\nu}$ of $A$, the induced representation
$i_{P}^{G}(\pi\boxtimes\chi)$ is irreducible. Here the parameter $\nu$ is in
$\mathfrak{a}^{*}$ and generic means outside a locally finite union of
hyperplanes which depends only on the infinitesimal character of $\pi$. Notice
that there is no other assumption on $\pi$ than being irreducible, so the
result is not limited to generalised principal series or standard
representations, for which the result is already well known.

##  1\. Introduction

Let $G$ be a real reductive group. We assume that there is a connected
algebraic reductive group $\mathbb{G}$ defined over $\mathbb{R}$ and that $G$
has finite index in $\mathbb{G}(\mathbb{R})$. Let $P$ be a parabolic subgroup
of $G$, with Langlands decomposition $P=MAN$, let $\pi$ be an irreducible
representation of $M$, and $\chi$ a character of $A$. Consider the induced
representation $\tau=i_{P}^{G}(\pi\otimes\chi)$ where $i_{P}^{G}$ is the
functor of (normalized) parabolic induction.

######  Theorem 1.1.

For generic $\chi$, the representation $\tau=i_{P}^{G}(\pi\otimes\chi)$ is
irreducible.

More precisely, since $A=\exp\mathfrak{a}_{0}$ where $\mathfrak{a}_{0}$ is a
real abelian Lie algebra, characters of $A$ are of the form

(1.1)
$\chi_{\nu}:\exp(X)\mapsto\chi_{\nu}(\exp(X))=\exp(\nu(X)),\quad(X\in\mathfrak{a}_{0})$

for some $\nu\in\mathfrak{a}^{*}$. “Generic ” in the Theorem means “for
$\chi=\chi_{\nu}$ with $\nu$ outside a countable locally finite union of
hyperplanes in $\mathfrak{a}^{*}$”. See (4.2) for the precise condition when
the infinitesimal character of $\pi$ is non singular and Section 5 for some
extra condition when infinitesimal character of $\pi$ is singular.

The result may seems obvious to experts, but I was not able to find a
reference in the literature. For $p$-adic groups, the analog result is well-
known, a proof is given by F. Sauvageot in [6] and a totally different one by
J-F. Dat [2]. Both proofs seems difficult to adapt to the real case. I propose
here a very simple argument, based on a very sophisticated theory, namely the
Kazhdan-Lusztig-Vogan theory of character multiplicities that I will now try
to describe (partly) in a few words.

Let $\mathcal{L}$ be a real reductive group, with Lie algebra
$\mathfrak{l}_{0}$ and maximal compact subgroup $\mathcal{K}$. Let
$\mathfrak{l}$ be the complexified Lie algebra of $\mathfrak{l}_{0}$. By
“representations of $\mathcal{L}$” we mean finite length Harish-Chandra
modules for the pair $(\mathfrak{l},\mathcal{K})$. Let us fix an infinitesimal
character $\chi$ for $\mathcal{L}$. Let us denote by
$\mathcal{H}\mathcal{C}(\mathfrak{l},\mathcal{K})_{\chi}$, or simply
$\mathcal{H}\mathcal{C}_{\chi}$, the category of representations of
$\mathcal{L}$ with infinitesimal character $\chi$, and by
$\mathbb{K}_{\mathbb{Z}}\mathcal{H}\mathcal{C}(\mathfrak{l},\mathcal{K})_{\xi}$,
or simply $\mathbb{K}_{\mathbb{Z}}\mathcal{H}\mathcal{C}_{\chi}$, its
Grothendieck group with coefficient in $\mathbb{Z}$. If $\pi$ is a
representation in $\mathcal{H}\mathcal{C}_{\chi}$, we denote by $[\pi]$ its
image in $\mathbb{K}_{\mathbb{Z}}\mathcal{H}\mathcal{C}_{\chi}$.

A result of Harish-Chandra asserts that the number of equivalence classes of
irreducible representations with fixed infinitesimal character is finite and
Langlands classification for irreducible representations, as reformulated by
Vogan (see [7], [8]), gives us a set
$\mathcal{P}_{\chi}^{\mathcal{L}}=\mathcal{P}_{\chi}$ which parametrizes the
equivalence classes of irreducible representations in
$\mathcal{H}\mathcal{C}_{\chi}$. I will be more precise later, but for the
moment, I will just say that a parameter $\gamma$ in $\mathcal{P}_{\chi}$ is
roughly a character of a Cartan subgroup of $\mathcal{L}$ with some additional
data, from which one can construct a “standard” representation
$\mathbf{std}(\gamma)$ in $\mathcal{H}\mathcal{C}_{\chi}$ (parabolically
induced from a limit of discrete series modulo the center of the corresponding
Levi subgroup). The standard representation $\mathbf{std}(\gamma)$ has a
Langlands quotient $\mathbf{irr}(\gamma)$ which is irreducible, and appears
with multiplicity one in $\mathbf{std}(\gamma)$. Thus,
$([\mathbf{irr}(\gamma)])_{\gamma\in\mathcal{P}_{\chi}}$ is a basis of the
Grothendieck group $\mathbb{K}_{\mathbb{Z}}\mathcal{H}\mathcal{C}_{\chi}$.
Therefore, in the Grothendieck group, one can write for all
$\delta\in\mathcal{P}_{\chi}$

(1.2)
$[\mathbf{std}(\delta)]=\sum_{\gamma\in\mathcal{P}_{\chi}}m(\gamma,\delta)\,[\mathbf{irr}(\gamma)]$

for some non negative integers $m(\gamma,\delta)$ (the multiplicity of
$\mathbf{irr}(\gamma)$ in $\mathbf{std}(\delta)$). Vogan has introduced a
length function $l_{I}$ on $\mathcal{P}_{\chi}$, such that if
$m(\gamma,\delta)\neq 0$, then $l_{I}(\gamma)<l_{I}(\delta)$, or
$\gamma=\delta$. Since $m(\gamma,\gamma)=1$, we see that one can invert the
linear system (1.2) and get

(1.3)
$[\mathbf{irr}(\delta)]=\sum_{\gamma\in\mathcal{P}_{\xi}}M(\gamma,\delta)\,[\mathbf{std}(\gamma)]$

for some integers $M(\gamma,\delta)$. The Kazhdan-Lusztig-Vogan theory gives
an algorithm to compute these integers $M(\gamma,\delta)$ (or equivalently the
$m(\gamma,\delta)$). We give details about the Kazhdan-Lusztig-Vogan algorithm
in Section 3.

Let us apply this theory to the problem of determining when a representation
$\tau=i_{P}^{G}(\pi\otimes\chi)$ as above is irreducible. For
$\nu\in\mathfrak{a}^{*}$, set $\pi_{\nu}=\pi\otimes\chi_{\nu}$, an irreducible
representation of $L=MA$, the Levi factor of $P$. From the previous paragraph
applied to $\mathcal{L}=L$ and $\chi$ the infinitesimal character of
$\pi_{\nu}$, we can write $\pi_{\nu}=\mathbf{irr}(\delta)$ for some parameter
$\delta\in\mathcal{P}_{\chi}^{L}$ and therefore

$[\pi_{\nu}]=[\mathbf{irr}(\delta)]=\sum_{\gamma\in\mathcal{P}_{\chi}^{L}}M^{L}(\gamma,\delta)\,[\mathbf{std}(\gamma)].$

By the exactness of the functor $i_{P}^{G}$, we get

$[\tau]=[i_{P}^{G}(\pi_{\nu})]=[i_{P}^{G}(\mathbf{irr}(\delta))]=\sum_{\gamma\in\mathcal{P}_{\chi}^{L}}M^{L}(\gamma,\delta)\,[i_{P}^{G}(\mathbf{std}(\gamma))].$

Now, the infinitesimal character $\chi$ for $L$ determines an infinitesimal
character for $G$ that we can still denote by $\chi$. Furthermore, a parameter
$\gamma\in\mathcal{P}_{\chi}^{L}$ can be extended to a parameter
$\gamma^{G}\in\mathcal{P}_{\chi}^{G}$ so that

$i_{P}^{G}(\mathbf{std}(\gamma))=\mathbf{std}(\gamma^{G}).$

Thus we get

$[\tau]=\sum_{\gamma\in\mathcal{P}_{\xi}^{L}}M^{L}(\gamma,\delta)\,[\mathbf{std}(\gamma^{G})]$

We can compare this to

(1.4)
$[\mathbf{irr}(\delta^{G})]=\sum_{\eta\in\mathcal{P}^{G}_{\xi}}M^{G}(\eta,\delta^{G})\,[\mathbf{std}(\eta)]$

to conclude that $\tau=\mathbf{irr}(\delta^{G})$ if the correspondence
$\gamma\mapsto\gamma^{G}$ from $\mathcal{P}_{\chi}^{L}$ to
$\mathcal{P}_{\chi}^{G}$ is bijective (in the level of conjugacy classes) and
furthermore $M^{L}(\gamma,\delta)=M^{G}(\gamma^{G},\delta^{G})$. We will see
first that this is the case if the infinitesimal character $\chi$ is non
singular for $L$ and its central part $\nu$ satisfies the condition (4.2)
(Thm. 4.2).

The idea is that the Kazhdan-Lusztig-Vogan (KLV for short) algorithm which
computes the multiplicities $M(\gamma,\delta)$ is determined by a set of data
attached to the parameters, and that this set of data is preserved under the
one-to-one correspondence $\gamma^{L}\mapsto\gamma^{G}$. To see this, and
explain how the KLV-algorithm works, we need to introduce a lot of structure
theory and results taken from Vogan’s papers (about integral root systems,
Cayley transforms, cross-actions, etc), which makes the paper a little bit
heavy, but the proofs consist mostly in careful bookkeeping.

In Section 2 and 3 of the paper, we introduce the material to be able to
describe the KLV algorithm (in case of non singular infinitesimal character).
The algorithm itself (what are the KLV-polynomials, how they give the
multiplicities $M(\gamma,\delta)$ and how to compute them) is explained at the
end of Section 3, and the main result for us here is that this algorithm is
completely determined by the structural data introduced in Section 2.6. In
section 4, we show that these data are “the same” for $L$ and $G$, if the
parameter $\nu$ introduced above satisfies the conditions (4.2). Indeed, we
see first that the Langlands-Vogan parameters sets $\mathcal{P}^{G}_{\chi}$
and $\mathcal{P}^{L}_{\chi}$ are then in one-to-one correspondence (in the
level of conjugacy classes), that the integral root systems attached to $\chi$
are the same in $L$ and $G$, and likewise for all the data in Section 2.6. In
the last section, we show how to extend the result to the case of singular
infinitesimal character, using the “translation data” in Chapter 16 of [1].

I would like to thank Nadir Matringe for asking me a reference for the generic
irreducibility of parabolic induction in the real case. I first thought this
was probably known and written somewhere, but I couldn’t find a reference.
After this note was written, David Vogan informed me that he knew the proof
given here (which is not surprising since this proof consists mainly in giving
references to his work) and also a more elementary one (in the sense that it
uses less sophisticated results on Lie algebra cohomology) but probably not
shorter to expose from the published results. I would also like to thank him.

I had thought, previously to N. Matringe’s question, of using this argument
with the KLV-algorithm to prove irreducibility of parabolic induction. Indeed,
in a work with Colette Moeglin on the construction of Arthur packets for real
classical groups [5], the last step is the construction of arbitrary packets
from packets of “good parity” on a maximal Levi subgroup by parabolic
induction. In ibid, Thm 4.4, it is stated without proof that this induction
preserves irreducibility, and I had at that time the vague idea of proving
this using the result established here. It turns out that Colette Moeglin was
faster in finding a (quite different, very clever and very difficult) proof,
which is in [4], Thm. 5.4.

This is a first version of the paper, written somehow quickly to provide a
reference for the main result. I hope to improve it in a next version. First
the exposition of the KLV-algorithm can probably be streamlined a bit.
Secondly, the extra assumption added in the case of singular infinitesimal
character may probably be relaxed in some way. Thirdly, I would like to expand
on some applications of the main result, like the one alluded above on our
construction of Arthur packets.

##  2\. Notation and preliminaries

For any real Lie algebra $\mathfrak{b}_{0}$, we denote by $\mathfrak{b}$ its
complexification. Let $G$ be a real reductive group as in the introduction.
Let $\mathfrak{g}_{0}$ be the Lie algebra of $G$. We also fix a Cartan
involution $\theta$ of $G$ with associated maximal compact subgroup $K$, and
associated Cartan decomposition
$\mathfrak{g}_{0}=\mathfrak{k}_{0}\oplus\mathfrak{p}_{0}$. We denote by
$\sigma$ the complex conjugation in $\mathfrak{g}$ relative to the real form
$\mathfrak{g}_{0}$. We fix an $G$-invariant non-degenerate symmetric bilinear
form $\langle.\,,.\rangle$ on $\mathfrak{g}$ (and $\mathfrak{g}^{*}$),
preserved by $\theta$ and which is positive definite on $\mathfrak{p}_{0}$ and
definite negative on $\mathfrak{k}_{0}$.

If a group $\mathcal{G}$ acts on a set $X$ and if $Y$ is a subset of $X$, we
denote by $\mathcal{G}^{Y}$ the centraliser of $Y$ in $\mathcal{G}$ and by
$\mathrm{Norm}_{\mathcal{G}}(Y)$ its normaliser and we use analogous notation
for a linear action of a Lie algebra.

###  2.1. Cartan subalgebras, Cartan subgroups, roots

We recall the following well-known facts about Cartan subgroups. A Cartan
subgroup $H$ is the centraliser in $G$ of a Cartan subalgebra
$\mathfrak{h}_{0}$ of $\mathfrak{g}_{0}$. If the Cartan subalgebra
$\mathfrak{h}_{0}$ is $\theta$-stable and decomposes as
$\mathfrak{h}_{0}=\mathfrak{t}_{0}\oplus\mathfrak{a}_{0}$, then the Cartan
subgroup $H$ decomposes as $H=TA$ (direct product) with $T=H\cap K$ and
$A=\exp\mathfrak{a}_{0}$. Let $\mathfrak{h}_{0}$ be a Cartan subalgebra of
$\mathfrak{g}_{0}$. Let us denote by $R(\mathfrak{g},\mathfrak{h})$ the root
system of $\mathfrak{h}$ in $\mathfrak{g}$, by $W(\mathfrak{g},\mathfrak{h})$
the corresponding complex Weyl group, and by $W(G,H)=\mathrm{Norm}_{G}(H)/H$
the real Weyl group. Depending on their values on $\mathfrak{h}_{0}$, roots
are classified as real, complex or imaginary. One can furthermore distinguish
between compact imaginary and non-compact imaginary roots (see [8] p. 150).
Let us denote by

$R_{\mathbb{R}}(\mathfrak{g},\mathfrak{h}),\quad
R_{i\mathbb{R}}(\mathfrak{g},\mathfrak{h}),\quad
R_{i\mathbb{R},c}(\mathfrak{g},\mathfrak{h}),\quad
R_{\mathbb{C}}(\mathfrak{g},\mathfrak{h})$

the subsets of $R(\mathfrak{g},\mathfrak{h})$ consisting in the real,
imaginary, compact and complex roots respectively. Let us denote by
$\check{\alpha}=2\frac{\alpha}{\langle\alpha,\alpha\rangle}\in\mathfrak{h}^{*}$
the coroot associated to a root $\alpha\in R(\mathfrak{g},\mathfrak{h})$ and
by $s_{\alpha}$ the reflection in $W(\mathfrak{g},\mathfrak{h})$ associated to
$\alpha$.

Via Harish-Chandra isomorphisms, any element $\lambda$ in the dual
$\mathfrak{h}^{*}$ of any Cartan subalgebra $\mathfrak{h}$ determines a
character $\chi_{\lambda}$ of $\mathfrak{Z}(\mathfrak{g})$, the center of the
enveloping algebra of $\mathfrak{g}$ (i.e. an infinitesimal character). Both
$\lambda$ and $\chi_{\lambda}$ are said to be non singular when
$\langle\alpha,\lambda\rangle\neq 0$ for all $\alpha\in
R(\mathfrak{g},\mathfrak{h})$. If a positive root system
$R^{+}(\mathfrak{g},\mathfrak{h})$ is given, $\lambda$ is said to be dominant
if $-\langle\check{\alpha},\lambda\rangle\notin\mathbb{N}$ for all $\alpha\in
R^{+}(\mathfrak{g},\mathfrak{h})$.

For $\lambda\in\mathfrak{h}^{*}$, set $R(\lambda)=\\{\alpha\in
R(\mathfrak{g},\mathfrak{h})\,|\,\langle\check{\alpha},\lambda\rangle\in\mathbb{Z}\\}$,
the set of integral roots (for $\lambda$) and let $W(\lambda)=W(R(\lambda))$
be the Weyl group of the root system $R(\lambda)$. If $\lambda$ is non-
singular, then put

$R^{+}(\lambda)=\\{\alpha\in
R(\lambda)\,|\,\langle\check{\alpha},\lambda\rangle>0\\}$

and let $\Pi(\lambda)$ and $S(\lambda)$ be respectively the set of simple
roots in $R^{+}(\lambda)$ and the set of simple reflection in $W(\lambda)$.

In order to be able to compare roots and Weyl groups on different Cartan
subalgebras, we will use the abstract Cartan subalgebra $\mathfrak{h}_{a}$ of
$\mathfrak{g}$ (see [8], Section 2). We fix a positive root system
$R^{+}(\mathfrak{g},\mathfrak{h}_{a})$ in $R(\mathfrak{g},\mathfrak{h}_{a})$
and a non singular dominant weight $\lambda_{a}\in\mathfrak{h}_{a}^{*}$. This
defines an infinitesimal character $\chi_{\lambda_{a}}$. We also define
$R^{a}=R(\lambda_{a})$, $R^{a,+}$, $W^{a}$, $\Pi^{a}$, $S^{a}$ to be
respectively the abstract integral root system, the abstract integral positive
root system, the abstract integral Weyl group, the abstract set of simple
roots, and the abstract set of simple reflections.

If $\mathfrak{h}$ is any Cartan subalgebra of $\mathfrak{g}$ and
$\lambda\in\mathfrak{h}^{*}$ is such that $\chi_{\lambda}=\chi_{\lambda_{a}}$,
there is an isomorphism
$i_{\lambda}:\mathfrak{h}_{a}^{*}\rightarrow\mathfrak{h}^{*}$ sending
$\lambda_{a}$ to $\lambda$ which induces isomorphisms $R^{a}\rightarrow
R(\lambda)$, $W^{a}\rightarrow W(\lambda)$, and so forth.

###  2.2. Parabolic subgroups

Let $P$ be a parabolic subgroup of $G$ with Langlands decomposition $P=MAN$
and Levi factor $L=MA$ (direct product). Denote by $\mathfrak{m}_{0}$,
$\mathfrak{a}_{0}$, $\mathfrak{n}_{0}$ and $\mathfrak{l}_{0}$ the respective
Lie algebras of $M$, $A$, $N$ and $L$. Conjugating with an element of $G$, we
may assume that $\mathfrak{l}_{0}$ is $\theta$-stable and $M_{K}:=L\cap
K=M\cap K$ is a maximal compact subgroup of $L$ and $M$. Both $L$ and $M$ are
in the class of groups defined in the introduction.

Let $\mathfrak{h}_{0}$ be a $\theta$-stable Cartan subalgebra of
$\mathfrak{l}_{0}$. It decomposes as

$\mathfrak{h}_{0}=\mathfrak{h}_{M,0}\oplus\mathfrak{a}_{0}=\mathfrak{t}_{0}\oplus\mathfrak{a}_{M,0}\oplus\mathfrak{a}_{0}.$

Let $R(\mathfrak{n},\mathfrak{h})$ be the set of roots in
$R(\mathfrak{g},\mathfrak{h})$ such that the corresponding root space is in
$\mathfrak{n}$. Then

$R(\mathfrak{g},\mathfrak{h})=R(\mathfrak{l},\mathfrak{h})\coprod
R(\mathfrak{n},\mathfrak{h})\coprod(-R(\mathfrak{n},\mathfrak{h})).$

The roots $\alpha\in R(\mathfrak{n},\mathfrak{h})$ are either real, or complex
with $\sigma(\alpha)=-\theta(\alpha)$ also in $R(\mathfrak{n},\mathfrak{h})$.
Let us choose a positive root system $R^{+}(\mathfrak{l},\mathfrak{h})$ and
set $R^{+}(\mathfrak{g},\mathfrak{h})=R^{+}(\mathfrak{l},\mathfrak{h})\coprod
R(\mathfrak{n},\mathfrak{h})$. By [3], Lemma 11.13 and (11.12) there is an
element $h_{\delta(\mathfrak{n})}\in\mathfrak{a}_{0}$ such that
$\mathfrak{l}_{0}=\mathfrak{g}_{0}^{h_{\delta(\mathfrak{n})}}$. Therefore, as
$\mathfrak{a}_{0}$ is central in $\mathfrak{l}_{0}$, we have
$\mathfrak{l}_{0}=\mathfrak{g}_{0}^{\mathfrak{a}_{0}}$.

Since $L=MA$ it is clear that $L\subset G^{\mathfrak{a}}_{0}=G^{A}$. If $g\in
G^{\mathfrak{a}}_{0}=G^{A}$, it preserves $\mathfrak{l}_{0}$,
$\mathfrak{n}_{0}$ and $\mathfrak{n}_{0}^{-}$ which are stable under the
adjoint action of $\mathfrak{a}_{0}$. Therefore
$L=G^{A}=G^{\mathfrak{a}_{0}}$.

Some parabolic subgroups called cuspidal are attached to Cartan subgroups :
let $H=TA$ be a $\theta$-stable Cartan subgroup. Set $L=G^{A}$ and $M=K^{A}$.
Then $L=MA$ is a Levi factor of parabolic subgroups $P=MAN$.

###  2.3. Cayley transforms

For the results in this section, we refer to [7], §8.3. Suppose that $H=TA$ is
a $\theta$-stable Cartan subgroup of $G$, and let $\alpha\in
R(\mathfrak{g},\mathfrak{h})$ be a real root. Then the root vectors for
$\alpha$ generate a subalgebra of $\mathfrak{g}_{0}$ isomorphic to
$\mathfrak{s}\mathfrak{l}(2,\mathbb{R})$ and we get a Lie algebra morphism

$\phi_{\alpha}:\,\mathfrak{s}\mathfrak{l}(2,\mathbb{R})\longrightarrow\mathfrak{g}_{0}$

satisfying $\phi_{\alpha}(-{}^{t}X)=\theta(X)$. We choose $\phi_{\alpha}$ so
that

$Z_{\alpha}:=\phi_{\alpha}\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}\in\mathfrak{a}_{0}\subset\mathfrak{h}_{0}\quad\text{and}\,\phi_{\alpha}\begin{pmatrix}0&1\\\
0&0\end{pmatrix}\in\mathfrak{g}_{0}^{\alpha}$

and set

$\tilde{Z}_{\alpha}:=\phi_{\alpha}\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}\in\mathfrak{k}_{0}.$

Since $G$ is linear, this map exponentiate to a group morphism :

(2.1) $\Phi_{\alpha}:\mathbf{SL}(2,\mathbb{R})\longrightarrow G.$

Put

$\sigma_{\alpha}=\Phi_{\alpha}\begin{pmatrix}0&1\\\ -1&0\end{pmatrix},\quad
m_{\alpha}=\sigma_{\alpha}^{2}.$

Then $m_{\alpha}\in T$ and $\sigma_{\alpha}\in K$. If $\alpha$ is real, then
$\sigma_{\alpha}$ normalizes $H$ and represents $s_{\alpha}\in W(G,H)$.

Define
$\mathfrak{h}_{0}^{\alpha}=\mathfrak{t}_{0}^{\alpha}\oplus\mathfrak{a}_{0}^{\alpha}$
by setting $\alpha_{0}^{\alpha}=\\{X\in\mathfrak{a}_{0}\,|\,\alpha(X)=0\\}$
and
$\mathfrak{t}_{0}^{\alpha}=\mathfrak{t}_{0}\oplus\mathbb{R}\widetilde{Z}_{\alpha}$.
The corresponding Cartan subgroup will be denoted
$c_{\alpha}(H)=H^{\alpha}=T^{\alpha}A^{\alpha}$. Notice that
$\sigma_{\alpha}\in T^{\alpha}$ and $m_{\alpha}\in T^{\alpha}\cap T$.

Let $\tilde{\alpha}=c_{\alpha}(\alpha)$ be the non compact imaginary root of
$\mathfrak{h}^{\alpha}$ in $\mathfrak{g}$ supported on
$\widetilde{Z}_{\alpha}$ (the Cayley transform of $\alpha$).

If $H_{1}=T_{1}A_{1}$ is a $\theta$-stable Cartan subgroup of $G$, and
$\beta\in R(\mathfrak{g},\mathfrak{h})$ is a non-compact imaginary root, one
can also define a Lie algebra morphism

$\phi_{\beta}:\,\mathfrak{s}\mathfrak{l}(2,\mathbb{R})\longrightarrow\mathfrak{g}_{0}$

which exponentiate to a group morphism

(2.2) $\Phi_{\beta}:\mathbf{SL}(2,\mathbb{R})\longrightarrow G,$

another Cartan subgroup
$c^{\beta}(H_{1})=H_{1}^{\beta}=T_{1}^{\beta}A_{1}^{\beta}$ and a real root
$\tilde{\beta}=c^{\beta}(\beta)\in R(\mathfrak{g},\mathfrak{h}_{1}^{\beta})$.
The two constructions are inverse of each other : if $H=TA$ is a
$\theta$-stable Cartan subgroup of $G$, and $\alpha\in
R(\mathfrak{g},\mathfrak{h})$ is a real root then
$\Phi_{\alpha}=\Phi_{\tilde{\alpha}}$, $c^{\tilde{\alpha}}(c_{\alpha}(H))=H$
and $c^{\tilde{\alpha}}(\tilde{\alpha})=\alpha$. If $H_{1}=T_{1}A_{1}$ is a
$\theta$-stable Cartan subgroup of $G$, and $\beta\in
R(\mathfrak{g},\mathfrak{h}_{1})$ is a non compact imaginary root then
$\Phi_{\beta}=\Phi_{\tilde{\beta}}$,
$c_{\tilde{\beta}}(c^{\beta}(H_{1}))=H_{1}$ and
$c_{\tilde{\beta}}(\tilde{\beta})=\beta$.

The following equivalent conditions define type I roots (for a real root
$\alpha$ or the corresponding non compact imaginary root $\tilde{\alpha}$) :

  * a)

the reflection $s_{\widetilde{\alpha}}$ does not belong to $W(G,H^{\alpha})$,

  * b)

$T_{1}^{\alpha}=T$,

  * c)

$\alpha:\,T\rightarrow\\{\pm 1\\}$ is not onto.

The following equivalent conditions define type II roots :

  * a)

the reflection $s_{\widetilde{\alpha}}$ belongs to $W(G,H^{\alpha})$,

  * b)

$T_{1}^{\alpha}$ has index $2$ in $T$ and $s_{\widetilde{\alpha}}$ has a
representative in $T\setminus T_{1}^{\alpha}$,

  * c)

$\alpha:\,T\rightarrow\\{\pm 1\\}$ is onto.

###  2.4. Parameters for Langlands classification

We start by fixing an infinitesimal character $\chi=\chi_{\xi_{a}}$ by picking
an element $\xi_{a}\in\mathfrak{a}_{a}^{*}$. We assume that $\xi_{a}$ is non
singular. Abstract integral roots, etc, defined in §2.1 with respect to an
element $\lambda_{a}\in\mathfrak{h}_{a}^{*}$ are now defined with respect to
this element $\xi_{a}$.

We recall the set of parameters ${\mathcal{P}^{G}_{\chi}}_{/\sim K}$ for the
Langlands classification introduced in [7] (see also [8]) of irreducible
representations of $G$ with infinitesimal character $\xi_{a}$.

######  Definition 2.1.

A parameter $\gamma$ is a multiplet

$\gamma=(H=TA,\Gamma,\bar{\gamma})$

where $H=TA$ is a $\theta$-stable Cartan subgroup of $G$, $\Gamma$ is a
character of $H$, and $\bar{\gamma}\in\mathfrak{h}^{*}$, satisfying the
following conditions $a)$, $b)$, $c)$:

  * a)

$\bar{\gamma}_{\mathfrak{t}}\in i\mathfrak{t}_{0}^{*}$, and
$\langle\alpha,\bar{\gamma}\rangle\neq 0$, $(\forall\alpha\in
R_{i\mathbb{R}}(\mathfrak{g},\mathfrak{h}))$.

Set

$\displaystyle
R^{+}_{i\mathbb{R}}=R^{+}_{i\mathbb{R}}(\mathfrak{g},\mathfrak{h})=\\{\alpha\in
R_{i\mathbb{R}}(\mathfrak{g},\mathfrak{h})|\,\langle\alpha,\bar{\gamma}\rangle>0\\},$
$\displaystyle R^{+}_{i\mathbb{R},c}=R^{+}_{i\mathbb{R}}\cap
R_{i\mathbb{R},c}(\mathfrak{g},\mathfrak{h})$
$\displaystyle\rho(R^{+}_{i\mathbb{R}})=\frac{1}{2}\sum_{\alpha\in
R^{+}_{i\mathbb{R}}}\alpha,$
$\displaystyle\rho(R^{+}_{i\mathbb{R},c})=\frac{1}{2}\sum_{\alpha\in
R^{+}_{i\mathbb{R},c}}\alpha.$

  * b)

$d\Gamma=\bar{\gamma}+\rho(R^{+}_{i\mathbb{R}})-2\rho(R^{+}_{i\mathbb{R},c})$

  * c)

The infinitesimal character $\chi_{\bar{\gamma}}$ equals $\chi$.

Attached to a parameter $\gamma=(H=TA,\Gamma,\bar{\gamma})$ as above, there is
a standard representation $\mathbf{std}(\gamma)$ (see [7] or [8]); it may be
defined by parabolic induction from a discrete series representation on a the
cuspidal parabolic subgroup $P=MAN$ attached to the Cartan subgroup $H=TA$.
The group $N$ is chosen so that the Langlands subquotients appear as quotients
of $\mathbf{std}(\gamma)$. For non singular infinitesimal character this
quotient is irreducible and is denoted by $\mathbf{irr}(\gamma)$.

Let us denote by $\mathcal{P}^{G}_{\chi}$ the set of parameter $\gamma$ as
above, and by ${\mathcal{P}^{G}_{\chi}}_{/\sim K}$ the $K$-conjugacy classes
in $\mathcal{P}^{G}_{\chi}$. The Langlands classication for non-singular
infinitesimal character $\chi$ is the following theorem (see [8] Thm. 2.13 and
the references given there).

######  Theorem 2.2.

Suppose that $\pi$ is an irreducible representation of $G$ with non-singular
infinitesimal character $\chi$. Then there is a parameter
$\gamma\in\mathcal{P}_{\chi}$ such that $\pi=\mathbf{irr}(\gamma)$. If two
parameters $\gamma_{1}$ and $\gamma_{2}$ satisfy
$\pi=\mathbf{irr}(\gamma_{1})=\mathbf{irr}(\gamma_{2})$, then $\gamma_{1}$ and
$\gamma_{2}$ are $K$-conjugate.

We will constantly abuse notation by denoting the $K$-conjugacy class of an
element $\gamma\in\mathcal{P}^{G}_{\chi}$ also by $\gamma$, and conversely,
for a conjugacy class $\gamma\in{\mathcal{P}^{G}_{\chi}}_{/\sim K}$, we denote
again by $\gamma$ the choice of a representative in $\mathcal{P}^{G}_{\chi}$.
Usually, this should not lead to any confusion.

###  2.5. Cayley transforms and cross-action on parameters

For any $\gamma=(H=TA,\Gamma,\bar{\gamma})\in{\mathcal{P}^{G}_{\chi}}$ and for
any $w\in W(\bar{\gamma})$, a new element $w\times\gamma$, with first
component $H=TA$ is defined in [7] §8.3. When $\alpha\in R^{+}(\bar{\gamma})$
is a simple root, the other components of $s_{\alpha}\times\gamma$ are given
explicitly in ibib. Lemma 8.3.2. One can use the isomorphisms
$i_{\bar{\gamma}}$ of §2.1 to transport this to an action of $W^{a}$ on
$\mathcal{P}^{G}_{\chi}$ and ${\mathcal{P}^{G}_{\chi}}_{/\sim K}$.

In Section 2.3, the Cayley transform of a $\theta$-stable Cartan subgroup
$H=TA$ with respect to a real root $\alpha$ has been recalled. In ibib. §8.3
this definition is extended to Langlands parameters.

We recall first the parity conditions on real integral roots. If
$\gamma=(H=TA,\Gamma,\bar{\gamma})\in{\mathcal{P}^{G}_{\chi}}$ and if
$\alpha\in R(\bar{\gamma})$ is a real root, we say that $\alpha$ satisfies the
parity condition with respect to $\gamma$ if

(2.3)
$\Gamma(m_{\alpha})=\epsilon^{G}_{\alpha}(-1)^{\langle\check{\alpha},\bar{\gamma}\rangle}.$

Here $\epsilon^{G}_{\alpha}\in\\{\pm\\}$ is defined in [7], Def. 8.3.11.

If $\alpha$ is a real integral root satisfying the parity condition, then the
Cayley transform $c_{\alpha}(\gamma)$ is defined as a subset of
$\mathcal{P}^{G}_{\xi}$. It is a singleton if $\alpha$ is type II, and we set
$c_{\alpha}(\gamma)=\\{\gamma_{\alpha}\\}$, and if $\alpha$ is type I, then
$c_{\alpha}(\gamma)=\\{\gamma_{\alpha}^{+},\gamma_{\alpha}^{-}\\}$, with
$s_{\tilde{\alpha}}\times\gamma_{\alpha}^{\pm}=\gamma_{\alpha}^{\mp}$. The
first component of $\gamma_{\alpha}$ or $\gamma_{\alpha}^{\pm}$ is
$H^{\alpha}=c_{\alpha}(H)$.

In the other direction, one can define also Cayley transform of a parameter
$\gamma=(H=TA,\Gamma,\bar{\gamma})\in{\mathcal{P}^{G}_{\chi}}$ with respect to
a non compact imaginary root $\beta\in R(\bar{\gamma})$. The Cayley transform
$c^{\beta}(\gamma)$ is a singleton if $\alpha$ is type I, and we set
$c^{\beta}(\gamma)=\\{\gamma^{\beta}\\}$, and if $\beta$ is type II, then
$c^{\beta}(\gamma)=\\{\gamma^{\beta}_{+},\gamma^{\beta}_{-}\\}$, with
$s_{\tilde{\beta}}\times\gamma^{\beta}_{\pm}=\gamma^{\beta}_{\mp}$. The first
component of $\gamma^{\beta}$ or $\gamma^{\beta}_{\pm}$ is
$H^{\beta}=c^{\beta}(H)$.

###  2.6. Data associated to a Langlands parameter

We associate to any
$\gamma=(H=TA,\Gamma,\bar{\gamma})\in{\mathcal{P}^{G}_{\chi}}$ the following
set of data :

  * (1)

$R(\bar{\gamma})$, the integral root system defined by $\bar{\gamma}$ and
$R^{+}(\bar{\gamma})$ the set of integral positive roots.

  * (2)

$R^{+}_{i\mathbb{R}}(\bar{\gamma})=R^{+}(\bar{\gamma})\cap
R^{+}_{i\mathbb{R}}(\mathfrak{g},\mathfrak{h})$, the set of integral imaginary
positive roots.

  * (3)

$R^{+}_{i\mathbb{R},c}(\bar{\gamma})=R^{+}(\bar{\gamma})\cap
R_{i\mathbb{R},c}(\mathfrak{g},\mathfrak{h})$, the set of integral imaginary
compact positive roots and
$R^{+}_{i\mathbb{R},nc}(\bar{\gamma})=R^{+}(\bar{\gamma})\cap
R_{i\mathbb{R},nc}(\mathfrak{g},\mathfrak{h})$, the set of integral imaginary
non compact positive roots.

  * (4)

$R^{+,I}_{i\mathbb{R},nc}(\bar{\gamma})$ the set of integral imaginary non
compact positive roots of type I and $R^{+,II}_{i\mathbb{R},nc}(\bar{\gamma})$
the set of integral imaginary non compact positive roots of type II.

  * (5)

$R_{\mathbb{R}}(\bar{\gamma})=R(\bar{\gamma})\cap
R_{\mathbb{R}}(\mathfrak{g},\mathfrak{h})$, the set of integral real roots and
$R^{+}_{\mathbb{R}}(\bar{\gamma})=R^{+}(\bar{\gamma})\cap
R_{\mathbb{R}}(\mathfrak{g},\mathfrak{h})$, the set of integral real positive
roots.

  * (6)

$R_{\mathbb{R},0}(\bar{\gamma})$, the set of integral real roots not
satisfying the parity condition for $\gamma$ and
$R_{\mathbb{R},1}(\bar{\gamma})$ is the set of integral real roots satisfying
the parity condition for $\gamma$.

  * (7)

$R^{I}_{\mathbb{R},1}(\bar{\gamma})$, the set of integral real roots
satisfying the parity condition for $\gamma$ of type I and
$R^{II}_{\mathbb{R},1}(\bar{\gamma})$ the set of integral real roots
satisfying the parity condition for $\gamma$ of type II.

  * (8)

$R^{+}_{\mathbb{C},1}(\bar{\gamma})$, the set of integral complex positive
roots such that $\theta(\alpha)\in R^{+}_{\mathbb{C}}(\bar{\gamma})$ and
$R^{+}_{\mathbb{C},0}(\bar{\gamma})$, the set of integral positive roots such
that $\theta(\alpha)\notin R^{+}_{\mathbb{C}}(\bar{\gamma})$

######  Remarks 2.3.

  * a)

The integral root system $R(\bar{\gamma})$ is $\theta$-stable ([7], Lemma
8.2.5).

  * b)

A parameter $\gamma\in\mathcal{P}^{G}_{\chi}$ is said to be minimal if
$R_{\mathbb{R},1}(\bar{\gamma})$ and $R^{+}_{\mathbb{R},0}(\bar{\gamma})$ are
empty. It is equivalent to the fact that for any non real simple root in
$R^{+}(\bar{\gamma})$, $\theta(\alpha)\in R^{+}(\bar{\gamma})$ and no real
simple root in $R^{+}(\bar{\gamma})$ satisfies the parity condition (see [7]
Def. 8.6.5). The standard representation $\mathbf{std}(\gamma)$ is irreducible
if and only if $\gamma$ is minimal ([7] Thm. 8.6.6).

  * c)

As we will see in the next section, the data above for all
$\gamma\in{\mathcal{P}^{G}_{\chi}}_{/\sim K}$ (a finite set) is sufficient to
determine the multiplicities $M(\gamma,\delta)$ or $m(\gamma,\delta)$ for any
$\gamma,\delta\in{\mathcal{P}^{G}_{\chi}}_{/\sim K}$ via the KLV-algorithm.

##  3\. The KLV-algorithm

We follow here [8], Section 12.

###  3.1. Blocks

To describe (part of) the KLV-algorithm, we first introduce the notion of
block in ${\mathcal{P}_{\chi}^{G}}_{/\sim K}$. Blocks are equivalence classes
on ${\mathcal{P}^{G}_{\chi}}_{/\sim K}$ for the equivalence relation generated
by

$\gamma_{1}\sim\gamma_{2}\text{ if }m(\gamma_{1},\gamma_{2})\neq 0,$

i.e. $\mathbf{irr}(\gamma_{1})$ occurs as a subquotient in
$\mathbf{std}(\gamma_{2})$. This is also the equivalence relation generated by
the weaker condition that $\gamma_{1}\sim\gamma_{2}$ if
$\mathbf{irr}(\gamma_{1})$ and $\mathbf{irr}(\gamma_{2})$ occur as subquotient
of the same standard representation $\mathbf{std}(\delta)$. Another
characterisation is given in ibid, Thm 9.2.11. Block equivalence is generated
by the following relations : if $\gamma\in\mathcal{P}^{G}_{\chi}$ and $\beta$
is a simple non-compact imaginary root in $R^{+}(\bar{\gamma})$, then
$\gamma\sim\gamma^{\prime}$ for any $\gamma^{\prime}\in c^{\beta}(\gamma)$,
and if $\alpha$ is a simple complex root in $R^{+}(\bar{\gamma})$, then
$\gamma\sim s_{\alpha}\times\gamma$.

Via the Langlands-Vogan parametrisation (Thm. 2.2), block equivalence gives an
equivalence relation on equivalence classes of irreducible representations
which can be characterized in terms of $\mathrm{Ext}$ groups (see [7] Def.
9.2.1 and Prop. 9.2.10). A result of Casselman (see [7], Cor. 9.2.24 states
that if two irreducible representations $\pi_{1}$ and $\pi_{2}$ of $G$ are in
different blocks, then $\mathrm{Ext}^{*}(\pi,\pi_{2})=0$. Therefore the set of
parameters $\mathcal{P}_{\xi}^{G}$ admits a partition in blocks

(3.1) ${\mathcal{P}^{G}_{\chi}}_{/\sim K}=\coprod_{i}\mathfrak{B}_{i}.$

Let us fix a block $\mathfrak{B}$ in the partition above.

######  Definition 3.1.

The integral length of a parameter
$\gamma=(H=TA,\Gamma,\bar{\gamma})\in\mathcal{P}^{G}_{\chi}$ is

$l_{I}(\gamma)=\frac{1}{2}\left|\\{\alpha\in
R^{+}(\bar{\gamma})|\,\theta(\alpha)\notin
R^{+}(\bar{\gamma})\\}\right|+\frac{1}{2}\dim A-c^{G}_{0}$

######  Remark 3.2.

The constant $c^{G}_{0}$ may be chosen so that $l_{I}(\gamma)\in\mathbb{N}$,
for all $\gamma\in\mathfrak{B}$, but the choice of $c_{0}^{G}$ is irrelevant
for the KLV-algorithm since it is always the difference
$l_{I}(\gamma_{1})-l_{I}(\gamma_{1})$ between the integral length of two
parameters $\gamma_{1},\,\gamma_{2}\in\mathcal{P}^{G}_{\chi}$ which matters.

The Hecke algebra $\mathcal{H}$ of the abstract Weyl group $W^{a}$ is defined
in [8], Def. 12.4. This is an algebra over
$\mathbb{Z}[u^{\frac{1}{2}},u^{-\frac{1}{2}}]$ generated by elements $T_{w}$,
$w\in W^{a}$ with the relations given in ibid.

The Hecke module of $\mathfrak{B}$ is the free module over
$\mathbb{Z}[u^{\frac{1}{2}},u^{-\frac{1}{2}}]$ with basis
$\\{\gamma\in\mathfrak{B}\\}$. Let us denote this module by
$\mathfrak{M}(\mathfrak{B})$. The action of $\mathcal{H}$ on
$\mathbb{Z}[u^{\frac{1}{2}},u^{-\frac{1}{2}}]$ is given also in ibid. More
precisely, what is given are formulas for $T_{s}\gamma$ when
$\gamma\in\mathfrak{B}$ and $s\in S^{a}$ is a simple reflection. This simple
reflection corresponds to a simple root $\alpha\in R^{+}(\bar{\gamma})$ via
the isomorphisms $i_{\bar{\gamma}}$ and the formula depends on $\alpha$. For
instance, if $\alpha$ is type II real satisfying the parity condition, then

$T_{s}\gamma=(u-1)\gamma-s_{\alpha}\times\gamma+(u-1)c_{\alpha}(\gamma).$

Let $s$ be a simple reflection in $S^{a}$. For any $\gamma_{1}$,
$\gamma_{2}\in\mathfrak{B}$, write $\gamma_{1}\stackrel{{\scriptstyle
s}}{{\rightarrow}}\gamma_{2}$ if $\alpha_{1}$, the corresponding simple root
in $R^{+}(\bar{\gamma}_{1})$ is complex with $\theta\alpha_{1}\notin
R^{+}(\bar{\gamma}_{1})$ and $\gamma_{2}=s\times\gamma_{1}$ or if $\alpha_{1}$
is real, satisfying the parity condition with respect to $\gamma_{1}$ and
$\gamma_{2}\in c_{\alpha_{1}}(\gamma_{1})$. Equivalently, if $\alpha_{2}$ is
the corresponding simple root in $R^{+}(\bar{\gamma}_{2})$, then $\alpha_{2}$
is complex and $\theta\alpha_{2}\notin R^{+}(\bar{\gamma}_{2})$ or
$\alpha_{2}$ is non compact imaginary with respect to $\gamma_{2}$ and
$\gamma_{1}\in c^{\alpha_{2}}(\gamma_{2})$.

If $\gamma_{1}\stackrel{{\scriptstyle s}}{{\rightarrow}}\gamma_{2}$, then
$l_{I}(\gamma_{2})=l_{I}(\gamma_{1})-1$, and we have also

$s\times\gamma_{1}\stackrel{{\scriptstyle
s}}{{\rightarrow}}s\times\gamma_{2},\quad\gamma_{1}\stackrel{{\scriptstyle
s}}{{\rightarrow}}s\times\gamma_{2},\quad
s\times\gamma_{1}\stackrel{{\scriptstyle s}}{{\rightarrow}}\gamma_{2}$

In [8], Def. 12.12 an order relation is defined on $\mathcal{B}$ and denoted
$\gamma_{1}\leq_{r}\gamma_{2}$. Let us just recall some properties of this
order relation.

  * a)

If $\gamma_{1}\leq_{r}\gamma_{2}$, then $l_{I}(\gamma_{1})\leq
l_{I}(\gamma_{2})$ and if $\gamma_{1}\leq_{r}\gamma_{2}$, and
$l_{I}(\gamma_{1})=l_{I}(\gamma_{2})$ then $\gamma_{1}=\gamma_{2}$.

  * b)

If $m(\gamma_{1},\gamma_{2})\neq 0$, or $M(\gamma_{1},\gamma_{2})$, then
$\gamma_{1}\leq_{r}\gamma_{2}$.

The next lemma is [8], Lemma 12.18

######  Lemma 3.3.

Suppose that $\gamma,\delta\in\mathfrak{B}$ and $m(\gamma,\delta)\neq 0$. Then
we can find $\delta^{\prime}\in\mathcal{B}$ and a simple reflection $s\in
W^{a}$ such that $\delta\stackrel{{\scriptstyle
s}}{{\rightarrow}}\delta^{\prime}$, and for any such $s$, one of the following
condition is satisfied

  * (i)

$m(\gamma,\delta^{\prime})\neq 0$.

  * (ii)

There exists $\gamma^{\prime}\in\mathfrak{B}$ such that
$\gamma\stackrel{{\scriptstyle s}}{{\rightarrow}}\gamma^{\prime}$ and
$m(\gamma^{\prime},\delta^{\prime})\neq 0$.

  * (iii)

Let $\alpha$ be the simple root in $R^{+}(\bar{\delta})$ corresponding to $s$.
Then $\alpha$ is real, satisfying the parity condition with respect to
$\delta$ and $(i)$ or $(ii)$ holds with $s\times\delta^{\prime}$ replacing
$\delta^{\prime}$.

The next ingredient in the KLV-algorithm is the duality map $D$ on
$\mathfrak{M}(\mathfrak{B})$ ([8], Lemma 12.14.

######  Lemma 3.4.

There is a unique $\mathbb{Z}$ linear map
$D:\mathfrak{M}(\mathfrak{B})\rightarrow\mathfrak{M}(\mathfrak{B})$ with the
following properties. For any $\gamma\in\mathfrak{B}$, write

$D\gamma=u^{-l_{I}(\gamma)}\sum_{\phi\in\mathfrak{B}}(-1)^{l_{I}(\gamma)-l_{I}(\phi)}R_{\phi\gamma}\,\phi$

for some polynomials
$R_{\phi\gamma}\in\mathbb{Z}[u^{\frac{1}{2}},u^{-\frac{1}{2}}]$. Then

  * a)

$D(um)=u^{-1}D(m),\;(\forall m\in\mathfrak{M}(\mathfrak{B}))$.

  * b)

$D((T_{s}+1)m)=u^{-1}(T_{s}+1)D(m),\;(\forall
m\in\mathfrak{M}(\mathfrak{B}),\,\forall s\in S^{a})$.

  * c)

$R_{\gamma\gamma}=1,\,(\forall\gamma\in\mathfrak{B})$.

  * d)

$R_{\phi\gamma}\neq
0\implies\phi\leq_{r}\gamma,\,(\forall\gamma\in\mathfrak{B})$.

The map $D$ has the following extra properties

  * e)

$R_{\phi\gamma}$ is a polynomial in $u$ of degree $\leq
l_{I}(\gamma)-l_{I}(\phi)$.

  * f)

$D^{2}=\mathrm{Id}_{\mathfrak{M}(\mathfrak{B})}$.

  * g)

The specialization of $D$ at $u=1$ is the identity.

We can finally define the KLV polynomials. This is [8], Lemma 12.15.

######  Lemma 3.5.

For any $\gamma\in\mathfrak{B}$, there is a unique element
$C_{\gamma}=\sum_{\phi\in\mathfrak{B}}P_{\phi\gamma}\phi\in\mathfrak{M}(\mathfrak{B})$
(with coefficients $P_{\phi\gamma}$ in
$\mathbb{Z}[u^{\frac{1}{2}},u^{-\frac{1}{2}}]$) such that
$D(C_{\gamma})=u^{-l_{I}(\gamma)}C_{\gamma}$, $P_{\gamma\gamma}=1$,
$P_{\phi\gamma}\neq 0$ only if $\phi\leq_{r}\gamma$ and if $\phi\neq\gamma$,
then $P_{\phi\gamma}$ is a polynomial in $u$ of degree
$\leq\frac{1}{2}\left(l_{I}(\gamma)-l_{I}(\phi)-1\right)$.

The next result is the Kazhdan-Lusztig-Vogan conjecture on multiplicities.

######  Theorem 3.6.

The integers $M(\gamma,\delta)$, $\gamma,\delta\in\mathfrak{b}$ are then given
by

$M(\gamma,\delta)=(-1)^{l_{I}(\delta)-l_{I}(\gamma)}P_{\gamma\delta}(1).$

It is proved by Vogan, Lusztig-Vogan if the infinitesimal character is
integral, and an argument by Bernstein settle the non-integral case. See [8]
and [1] for a discussion of this fundamental result and references to the
original papers.

Finally, there is an algorithm which computes the KLV-polynomials
$P_{\gamma,\delta}$. It is described in Prop. 6.14 of [9]. It starts with the
fact that $P_{\delta,\delta}=1$ for any
$\delta\in{\mathcal{P}_{\xi}^{G}}_{/\sim K}$ and that $P_{\gamma\delta}=0$ if
$\gamma\nleq_{r}\delta$ in ${\mathcal{P}_{\xi}^{G}}_{/\sim K}$. If
$P_{\gamma^{\prime},\delta^{\prime}}$ is known whenever
$l_{I}(\delta^{\prime})<l_{I}(\delta)$ or
$l_{I}(\delta^{\prime})=l_{I}(\delta)$ and
$l_{I}(\gamma^{\prime})>l_{I}(\gamma)$, then there are formulas for computing
$P_{\gamma\delta}=0$. What is important for us here is that to apply these
formulas, we only need the data associated to all
$\gamma\in{\mathcal{P}_{\xi}^{G}}_{/\sim K}$ in Section 2.6.

##  4\. Data in $G$ versus data in $L^{\flat}$

Let lus fix now a parabolic subgroup $P^{\flat}=M^{\flat}A^{\flat}N^{\flat}$
of $G$ with $\theta$-stable Levi factor $L^{\flat}=M^{\flat}A^{\flat}$. We
also fix a fundamental $\theta$-stable $H^{\flat}$ of $L$. Such a Cartan
subgroup has a decomposition $H^{\flat}=T^{\flat}A_{M^{\flat}}A^{\flat}$. Of
course there are similar decompositions for the Cartan subalgebras.

All the notation and results in Section 2 apply to the group $L^{\flat}$
instead of $G$. When needed, we will add a superscript $G$ or $L^{\flat}$ to
distinguish between objects defined relatively to $G$ or $L^{\flat}$.

Cartan subalgebras of $\mathfrak{l}_{0}^{\flat}$ are Cartan subalgebras of
$\mathfrak{g}_{0}$ and Cartan subgroups of $L^{\flat}$ are Cartan subgroups of
$G$. This is particular the case for $\mathfrak{h}_{0}^{\flat}$ and
$H^{\flat}=T^{\flat}A_{M^{\flat}}^{\flat}A^{\flat}$.

In general, a Cartan subgroup $H$ in $L^{\flat}$ decomposes as
$H=H_{M^{\flat}}A^{\flat}$ with $H_{M^{\flat}}$ a Cartan subgroup of
$M^{\flat}$. If $H$ is $\theta$-stable, one can further decompose
$H_{M^{\flat}}$ as $H_{M^{\flat}}=TA_{M^{\flat}}$ and $H$ as
$H=TA_{M^{\flat}}A^{\flat}$. For such a Cartan subgroup, we have as in Section
2.2

(4.1)
$R(\mathfrak{g},\mathfrak{h})=R(\mathfrak{l}^{\flat},\mathfrak{h})\coprod
R(\mathfrak{n}^{\flat},\mathfrak{h})\coprod(-R(\mathfrak{n}^{\flat},\mathfrak{h}))$

The roots $\alpha\in R(\mathfrak{n}^{\flat},\mathfrak{h})$ are either real, or
complex with $\sigma(\alpha)=-\theta(\alpha)$ also in
$R(\mathfrak{n}^{\flat},\mathfrak{h})$. Therefore

$R_{i\mathbb{R}}(\mathfrak{g},\mathfrak{h})=R_{i\mathbb{R}}(\mathfrak{l}^{\flat},\mathfrak{h}),\quad
R_{i\mathbb{R},c}(\mathfrak{g},\mathfrak{h})=R_{i\mathbb{R},c}(\mathfrak{l}^{\flat},\mathfrak{h})$

We can therefore simply write $R_{i\mathbb{R}}$ and $R_{i\mathbb{R},c}$ for
these system of imaginary roots.

######  Proposition 4.1.

The following conjugacy classes of Cartan subgroups are in natural one-to-one
correspondences. On one hand :

  * a)

$G$-conjugacy classes of Cartan subgroups of $G$ containing a $G$-conjugate of
$A^{\flat}$.

  * b)

$K$-conjugacy classes of $\theta$-stable Cartan subgroups of $G$ containing a
$G$-conjugate of $A^{\flat}$.

On the other hand :

  * c)

$L^{\flat}$-conjugacy classes of Cartan subgroups of $L^{\flat}$.

  * d)

$M_{K}^{\flat}$-conjugacy classes of $\theta$-stable Cartan subgroups of
$L^{\flat}$.

  * e)

$M^{\flat}$-conjugacy classes of Cartan subgroups of $M^{\flat}$.

  * f)

$M^{\flat}_{K}$-conjugacy classes of $\theta$-stable Cartan subgroups of
$M^{\flat}$.

Furthermore, the natural map from the set of conjugacy classes in $c)$ to the
set of conjugacy classes in $a)$ is surjective. The $G$-conjugacy classes in
$a)$ are exactly the ones which are greater than $H^{\flat}$ in the Hirai
order for $G$. In particular, it contains the maximally split $G$-conjugacy
class of Cartan subgroups of $G$.

Proof. The equivalence of $a)$ and $b)$ is in [7], Lemma 0.1.6. Of course, it
gives also the equivalence between $e)$ and $f)$ and $c)$ and $d)$. The
equivalence of $c)$ and $e)$ is clear since $A^{\flat}$ is central in
$L^{\flat}$. A Cartan subgroup containing $A^{\flat}$ is contained in
$G^{A^{\flat}}=L^{\flat}$ since Cartan subgroups are abelian by assumption,
proving that the natural map from the set of conjugacy classes in $c)$ to the
set of conjugacy classes in $a)$ is surjective. This map respects the Bruhat
order (for $L^{\flat}$ and $G$ respectively), and from this we get that the
set of conjugacy classes in $a)$ are greater than the one of $H^{\flat}$ in
the Bruhat order for $G$. Conversely, a Cartan subgroup of $G$ with
$G$-conjugacy class greater than the one of $H^{\flat}$ in the Bruhat order
for $G$ has a $G$-conjugate containing $A^{\flat}$. If two Cartan subgroups of
$L^{\flat}$ are $L^{\flat}$-conjugate, they are $G$-conjugate. In general, two
$G$-conjugate Cartan subgroups of $L^{\flat}$ are not $L^{\flat}$-conjugate,
unless they are maximally compact or split in $L^{\flat}$. ∎

We now fix an infinitesimal character $\chi=\chi_{\xi}$, both for $G$ and
$L^{\flat}$, by choosing $\xi\in(\mathfrak{h}^{\flat})^{*}$. We decompose
$\xi$ as $\xi=\xi_{M^{\flat}}+\nu$, according to the decomposition
$(\mathfrak{h}^{\flat})^{*}=(\mathfrak{h}_{M^{\flat}}^{\flat})^{*}\oplus(\mathfrak{a}^{\flat})^{*}$.
We assume that $\xi$ is non-singular for $L^{\flat}$, i.e.
$\langle\xi_{M^{\flat}},\check{\alpha}\rangle\neq 0$ for $\alpha\in
R(\mathfrak{l}^{\flat},\mathfrak{h}^{\flat})$.

Consider the following condition on $\nu\in\mathfrak{a}^{*}$.

(4.2) $\langle\check{\alpha},\xi_{M^{\flat}}+\nu\rangle\notin\mathbb{Z},\text{
and }\langle\check{\alpha},\nu\rangle\neq 0\quad(\forall\alpha\in
R(\mathfrak{n},\mathfrak{h}))$

If $\nu$ satisfies this condition, we have

(4.3) $R^{G}(\xi_{M^{\flat}}+\nu)=R^{L^{\flat}}(\xi_{M^{\flat}}+\nu)$

Furthermore $\xi$ is non singular also for $G$ and
$\mathfrak{g}^{\nu}=\mathfrak{l}$. Since $L(\mathbb{C})$ is connected
$G(\mathbb{C})^{\nu}=L^{\flat}(\mathbb{C})$.

Our main result is

######  Theorem 4.2.

Let $\pi$ be an irreducible representation of $M^{\flat}$. Then for any
$\nu\in(\mathfrak{a}^{\flat})^{*}$ satisfying (4.2), the induced
representation $i_{P^{\flat}}^{G}(\pi\boxtimes\chi_{\nu})$ is irreducible.

We will prove this theorem following the ideas given in the introduction. We
start by comparing the parameters for irreducible representations with
infinitesimal character $\chi=\chi_{\xi}$, for $G$ and $L^{\flat}$. To be
coherent with our preceding notation, we also fix a dominant $\xi_{a}$ in the
dual of the abstract Cartan subalgebra $\mathfrak{h}_{a}$ such that
$\chi_{\xi_{a}}=\chi_{\xi}=\chi$.

Consider a parameter
$\gamma^{L^{\flat}}=(H=TA,\Gamma,\bar{\gamma})\in\mathcal{P}^{L^{b}}_{\chi}$
as in Section 2.4 (but for $L^{\flat}$). Since imaginary roots are the same
for $\mathfrak{l}^{\flat}$ and $\mathfrak{g}$, it is clear that it is also a
parameter in $\mathcal{P}^{G}_{\chi}$ and conversely (see $(2)$ and $(3)$
below). So the identity map

$\mathcal{P}^{L^{\flat}}_{\chi}\longrightarrow\mathcal{P}^{G}_{\chi},\quad\gamma^{L}\mapsto\gamma^{G},$

which induces a surjective map

(4.4) ${\mathcal{P}^{L^{\flat}}_{\chi}}_{/\sim
M_{K}^{\flat}}\longrightarrow{\mathcal{P}^{G}_{\chi}}_{/\sim
K},\quad\gamma^{L}\mapsto\gamma^{G},$

######  Proposition 4.3.

The map (4.4) is one-to-one.

Proof. We have to show that two parameters
$\gamma_{1}=(H_{1}=T_{1}A_{1},\Gamma_{1},\bar{\gamma}_{1})$ and
$\gamma_{2}=(H_{2}=T_{2}A_{2},\Gamma_{2},\bar{\gamma}_{2})\in\mathcal{P}^{L^{b}}_{\chi}$
are $G$-conjugate, then they are $L^{\flat}$-conjugate. Since
$\bar{\gamma}_{1}$ and $\bar{\gamma}_{2}$ define the same infinitesimal
character as $\xi$, there are elements $l_{1}$ and $l_{2}$ in the complex
group $L^{\flat}(\mathbb{C})$ such that
$l_{1}\cdot\bar{\gamma}_{1}=\xi=l_{2}\cdot\bar{\gamma}_{2}$, and since
$\gamma_{1}$ and $\gamma_{2}$ are $G$-conjugate, there is an element $g\in G$
such that $g\cdot\bar{\gamma}_{1}=\bar{\gamma}_{2}$. Therefore, setting
$n=l_{2}gl_{1}^{-1}\in G(\mathbb{C})$, we get $n\cdot\xi=\xi$ with
$n\in\mathrm{Norm}_{G(\mathbb{C})}(\mathfrak{h}^{\flat})$. Since $\xi$ is non-
singular, we must have
$n\in\mathrm{Centr}_{G(\mathbb{C})}(\mathfrak{h}^{\flat})$. In particular
$n\cdot\nu=l_{2}gl_{1}^{-1}\cdot\nu$ so $g\cdot\nu=\nu$. Since by assumption
$G(\mathbb{C})^{\nu}=L^{\flat}(\mathbb{C})$, we get $g\in G\cap
L^{\flat}(\mathbb{C})=L^{\flat}$. ∎

We now check that the data $(1)$ to $(8)$ in Section 2.6 associated to
parameters are preserved by the correspondence
$\gamma^{L^{\flat}}\mapsto\gamma^{G}$ in (4.4). So, let us fix
$\gamma^{L^{\flat}}=(H=TA,\Gamma,\bar{\gamma})\in\mathcal{P}^{G}_{\chi}$. We
add superscript $G$ or $L^{\flat}$ to the various objects defined in Section
2.4 to distinguished the ones defined with respect to $G$ from the ones
defined with respect to $L^{\flat}$.

  * (1)

Because of condition (4.2) on $\nu\in(\mathfrak{h}^{\flat})^{*}$, we have
$R^{G}(\xi_{M^{\flat}}+\nu)=R^{L^{\flat}}(\xi_{M^{\flat}}+\nu)$ and therefore
$R^{L^{\flat}}(\bar{\gamma})=R^{G}(\bar{\gamma})$. Thus the integral root
systems for $\bar{\gamma}$ are the same for $L^{\flat}$ and $G$.

  * (2) and (3)

We have seen that roots in $R(\mathfrak{n}^{\flat},\mathfrak{h})$ are either
real, or complex with $\sigma(\alpha)=-\theta(\alpha)$. Therefore the
imaginary roots for $\mathfrak{h}$ are the same in $\mathfrak{l}^{\flat}$ and
$\mathfrak{g}$, and such an imaginary root is compact in $G$ if and only if it
is compact in $L^{\flat}$. Therefore

$R^{G}_{i\mathbb{R}}=R^{L^{\flat}}_{i\mathbb{R}},\quad
R^{G}_{i\mathbb{R},c}=R^{L^{\flat}}_{i\mathbb{R},c}\quad
R^{G}_{i\mathbb{R},nc}=R^{L^{\flat}}_{i\mathbb{R},nc}$
$R^{G,+}_{i\mathbb{R}}=R^{L^{\flat},+}_{i\mathbb{R}},\quad
R^{G,+}_{i\mathbb{R},c}=R^{L^{\flat},+}_{i\mathbb{R},c}\quad
R^{G,+}_{i\mathbb{R},nc}=R^{L^{\flat},+}_{i\mathbb{R},nc}$
$\rho(R^{G,+}_{i\mathbb{R}})=\rho(R^{L^{\flat},+}_{i\mathbb{R}}),\quad\rho(R^{G,+}_{i\mathbb{R},c})=\rho(R^{L^{\flat},+}_{i\mathbb{R},c})$

  * (4)

The fact that a non compact imaginary root $\tilde{\alpha}$ is type I or type
II in $G$ depends only on the map
$\Phi_{\alpha}:\mathbf{SL}(2,\mathbb{R})\longrightarrow G$ in (2.1) as it is
clear from the equivalent conditions defining type I or type II. Since in our
context we can choose the same map
$\Phi_{\alpha}:\mathbf{SL}(2,\mathbb{R})\longrightarrow L^{\flat}\subset G$
for $G$ and $L^{\flat}$ shows that a non compact imaginary root is type I in
$L^{\flat}$ if and only if it is type I in $G$.

  * (5)

Since $R^{G}(\bar{\gamma})=R^{L^{\flat}}(\bar{\gamma})$, we have also
$R_{\mathbb{R}}^{G}(\bar{\gamma})=R_{\mathbb{R}}^{L^{\flat}}(\bar{\gamma})$
and likewise
$R_{\mathbb{R}}^{G,+}(\bar{\gamma})=R_{\mathbb{R}}^{L^{\flat},+}(\bar{\gamma})$.

  * (6)

Let $\alpha\in R^{G}(\bar{\gamma})=R^{L^{\flat}}(\bar{\gamma})$ be an integral
real root. Since $m_{\alpha}$ is defined via the same map $\Phi_{\alpha}$ for
$L^{\flat}$ and $G$, we have only to check that
$\epsilon_{\alpha}^{L^{\flat}}=\epsilon_{\alpha}^{G}$. We do that in the lemma
below. Therefore the parity condition is the same in $L^{\flat}$ and $G$:

$R^{L^{\flat}}_{\mathbb{R},0}(\bar{\gamma})=R^{L^{\flat}}_{\mathbb{R},0}(\bar{\gamma})\text{
and
}R^{L^{\flat}}_{\mathbb{R},1}(\bar{\gamma})=R^{L^{\flat}}_{\mathbb{R},1}(\bar{\gamma}).$

  * (7)

As for non compact imaginary roots, real roots are of the same type (I or II)
with respect to $L^{\flat}$ and $G$.

  * (8)

Since $R^{G}(\bar{\gamma})=R^{L^{\flat}}(\bar{\gamma})$, we have also
$R_{\mathbb{C}}^{G}(\bar{\gamma})=R_{\mathbb{C}}^{L^{\flat}}(\bar{\gamma})$
and likewise
$R_{\mathbb{C}}^{G,+}(\bar{\gamma})=R_{\mathbb{C}}^{L^{\flat},+}(\bar{\gamma})$,
$R_{\mathbb{C},0}^{G,+}(\bar{\gamma})=R_{\mathbb{C},0}^{L^{\flat},+}(\bar{\gamma})$and
$R_{\mathbb{C}_{1}}^{G,+}(\bar{\gamma})=R_{\mathbb{C},1}^{L^{\flat},+}(\bar{\gamma})$.

######  Lemma 4.4.

For any integral real root $\alpha\in R_{\mathbb{R}}(\bar{\gamma})$,
$\epsilon_{\alpha}^{L^{\flat}}=\epsilon_{\alpha}^{G}$.

Proof. The sign $\epsilon_{\alpha}^{G}$ is defined as $(-1)^{d+1}$ where $d$
is an integer given by one of the definitions in [7] Lemma 8.3.9. For us, the
most convenient is the first one, i.e. we take $d=d_{1}$ in ibid. We fix
$\Phi_{\alpha}:\mathbf{SL}(2,\mathbb{R})\rightarrow L^{\flat}$ as in Section
2.3. Thus we get $m_{\alpha}$ and $H^{\alpha}=T^{\alpha}A^{\alpha}$. Consider
a cuspidal parabolic subgroup $P_{\alpha}=M_{\alpha}A_{\alpha}N_{\alpha}$
attached to $H^{\alpha}$, i.e.
$L^{\alpha}=M_{\alpha}A_{\alpha}=G^{A_{\alpha}}$. Up to conjugacy in
$L^{\flat}$, we may assume that $A^{\flat}\subset A^{\alpha}$ since
$H^{\alpha}$ is a Cartan subgroup of $L^{\flat}$, and therefore greater than
$H^{\flat}$ in the Hirai order. Thus $M_{\alpha}\subset M^{\flat}$. Therefore
the integer $d_{1}$ defined in [7] Lemma 8.3.9, which is
$d_{1}=\frac{1}{2}\dim\left((-1)-\text{ eigenspace of }m_{\alpha}\text{ in
}\mathfrak{m}_{\alpha}\cap\mathfrak{k}\right)$ equals

$\frac{1}{2}\dim\left((-1)-\text{ eigenspace of }m_{\alpha}\text{ in
}\mathfrak{m}_{\alpha}\cap\mathfrak{k}_{\mathfrak{l}^{\flat}}\right),$

since
$\mathfrak{m}_{\alpha}\cap\mathfrak{k}=\mathfrak{m}_{\alpha}\cap\mathfrak{m}^{\flat}\cap\mathfrak{k}=\mathfrak{m}_{\alpha}\cap\mathfrak{k}_{\mathfrak{l}^{\flat}}$.
Therefore
$\epsilon_{\alpha}^{L^{\flat}}=(-1)^{d_{1}+1}=\epsilon_{\alpha}^{G}$. ∎

Let us now consider the decomposition of the parameter sets
${\mathcal{P}^{L^{\flat}}_{\chi}}_{/\sim M_{K}^{\flat}}$ and
${\mathcal{P}^{G}_{\chi}}_{/\sim K}$ into blocks as in (3.1). From the
characterization of blocks in terms of Cayley transform and cross action, we
see that the bijective correspondence $\gamma^{L^{\flat}}\mapsto\gamma^{G}$
respects blocks. Furthermore, given a block $\mathfrak{B}^{L^{\flat}}$ in
${\mathcal{P}^{L^{\flat}}_{\chi}}_{/\sim M_{K}^{\flat}}$, we see that the
integral length (Def. 3.1) is the same for $\mathfrak{B}^{L^{\flat}}$ and the
corresponding block $\mathfrak{B}^{G}$, if we choose the constants $c_{0}^{G}$
and $c_{0}^{L^{\flat}}$ to be equal.

The bijective correspondence $\gamma^{L^{\flat}}\mapsto\gamma^{G}$ between
$\mathfrak{B}^{L^{\flat}}$ and $\mathfrak{B}^{G}$ also respects the relations
$\gamma\stackrel{{\scriptstyle s}}{{\rightarrow}}\delta$ since these relation
are determined by the data (1) to (8) of §2.6. The correspondence also respect
the Bruhat order $\leq_{r}$ since it can be characterised also in terms of
cross-action and Cayley transforms and the relations
$\gamma\stackrel{{\scriptstyle s}}{{\rightarrow}}\delta$ ([8], Lemma 12.13.

The Hecke algebra $\mathcal{H}$ is attached to the abstract integral Weyl
group $W^{a}$ which is the same for $L^{\flat}$ and $G$, since the integral
root systems are the same, $R^{G}(\xi)=R^{L^{\flat}}(\xi)$. The Hecke modules
$\mathfrak{M}(\mathfrak{B}^{l^{\flat}})$ and $\mathfrak{M}(\mathfrak{B}^{G})$
can also be identified as $\mathbb{Z}[u^{\frac{1}{2}},u^{-\frac{1}{2}}]$ via
the correspondence $\gamma^{L^{\flat}}\mapsto\gamma^{G}$ between
$\mathfrak{B}^{L^{\flat}}$ and $\mathfrak{B}^{G}$. The $\mathcal{H}$-module
structure is determined by the data data (1) to (8) of §2.6 which is also
preserved under the correspondence and therefore
$\mathfrak{M}(\mathfrak{B}^{L^{\flat}})$ and $\mathfrak{M}(\mathfrak{B}^{G})$
are isomorphic as $\mathcal{H}$-modules. As a consequence, the KLV-polynomials
$P_{\gamma\delta}$ are equal, i.e.

$P_{\gamma^{L^{\flat}}\delta^{L^{\flat}}}=P_{\gamma^{G}\delta^{G}},\quad(\forall\gamma^{L^{\flat}},\,\delta^{L^{\flat}}\in\mathfrak{B}^{L^{\flat}})$

As a corollary, we see that the multiplicities $M(\gamma,\delta)$ and
$m(\gamma,\delta)$ also coincide, i.e.

$M^{L^{\flat}}(\gamma^{L^{\flat}},\delta^{L^{\flat}})=M^{G}(\gamma^{G},\delta^{G}),\quad(\forall\gamma^{L^{\flat}},\,\delta^{L^{\flat}}\in\mathfrak{B}^{L^{\flat}})$

and

$m^{L^{\flat}}(\gamma^{L^{\flat}},\delta^{L^{\flat}})=m^{G}(\gamma^{G},\delta^{G}),\quad(\forall\gamma^{L^{\flat}},\,\delta^{L^{\flat}}\in\mathfrak{B}^{L^{\flat}}).$

As explained in the introduction this proves Thm. 4.2.∎

##  5\. The case of singular infinitesimal character

We turn now to the case of possibly singular infinitesimal character
$\xi\in{\mathfrak{h}^{\flat}}^{*})$. The relevant discussion may be found in
[1], Chapter 11. and [10]. First, the parameters have to be enriched by an
extra piece of data, so a parameter is now a multiplet

$\gamma=(H=TA,\Gamma,\bar{\gamma},R_{i\mathbb{R}}^{+})$

where $(H=TA,\Gamma,\bar{\gamma})$ is as before and $R_{i\mathbb{R}}^{+}$ is a
system of positive imaginary roots of $\mathfrak{h}$ in $\mathfrak{g}$. The
conditions imposed on $\gamma$ are the following $a)$, $b)$, $c)$, $d)$ and
$e)$ :

  * a)

$\langle\alpha,\bar{\gamma}\rangle\geq 0$, $(\forall\alpha\in
R_{i\mathbb{R}}^{+})$.

With $R^{+}_{i\mathbb{R},c}$, $\rho(R^{+}_{i\mathbb{R}})$ and
$\rho(R^{+}_{i\mathbb{R},c})$ as before, condition $b)$ and $c)$ are the same
as above for non-singular infinitesimal character.

  * d)

Suppose $\alpha$ is a simple root in $R_{i\mathbb{R}}^{+}$ such that
$\langle\alpha,\bar{\gamma}\rangle=0$. Then $\alpha$ is non compact.

  * e)

Suppose $\alpha$ is a real root in $R(\mathfrak{g},\mathfrak{h})$ such that
$\langle\alpha,\bar{\gamma}\rangle=0$. Then $\alpha$ does not satisfy the
parity condition, i.e. $\Gamma(m_{\alpha})=-\epsilon_{\alpha}^{G}$.

We want to show that Prop. 4.3 is still valid in this context, maybe by adding
some extra hypotheses on $\nu$. Starting the proof in the same way, with
$l_{1}\cdot\mathfrak{h}_{1}=\mathfrak{h}^{\flat}$,
$l_{1}\cdot\bar{\gamma}_{1}=\xi$,
$l_{2}\cdot\mathfrak{h}_{2}=\mathfrak{h}^{\flat}$,
$l_{2}\cdot\bar{\gamma}_{2}=\xi$, and
$g\cdot\mathfrak{h}_{1}=\mathfrak{h}_{2}$,
$g\cdot\bar{\gamma}_{1}=\bar{\gamma}_{2}$, we get to $n\cdot\xi=\xi$ that we
rewrite as $\xi_{M^{\flat}}-n\cdot\xi_{M^{\flat}}=n\cdot\nu-\nu$. We want to
impose some conditions on $\nu$ in order to deduce that $n\cdot\nu=\nu$, and
the rest of the proof will be the same. So if the linear map

$\phi_{n}:\,(\mathfrak{a}^{\flat})^{*}\longrightarrow(\mathfrak{h}^{\flat})^{*},\quad\nu\mapsto
n\cdot\nu-\nu$

is non zero, its kernel is a strict subspace of $(\mathfrak{a}^{\flat})^{*}$
and the solutions in $(\mathfrak{a}^{\flat})^{*}$ of
$n\cdot\nu-\nu=\xi_{M^{\flat}}-n\cdot\xi_{M^{\flat}}$ is strict affine
subspace. Since $\xi_{M^{\flat}}-n\cdot\xi_{M^{\flat}}$ takes only a finite
number of non-zero values for
$n\in\mathrm{Norm}_{\mathbb{G}(\mathbb{C})}(\mathfrak{h}^{\flat})$, we see
that for $\nu$ outside a finite number of strict affine subspaces in
$(\mathfrak{a}^{\flat})^{*}$,
$\xi_{M^{\flat}}-n\cdot\xi_{M^{\flat}}=n\cdot\nu-\nu$ implies $n\cdot\nu-\nu$.

Extra conditions on $\nu$. For any $w\in W(\mathfrak{g},\mathfrak{h}^{\flat})$
such that $\xi_{M^{\flat}}-w\cdot\xi_{M^{\flat}}$ is non-zero, $\nu$ is not in
the strict affine subspace of solutions of
$w\cdot\nu-\nu=\xi_{M^{\flat}}-w\cdot\xi_{M^{\flat}}$.

Under (4.2) and the extra conditions above, the map

(5.1) ${\mathcal{P}^{L^{\flat}}_{\chi}}_{/\sim
M_{K}^{\flat}}\longrightarrow{\mathcal{P}^{G}_{\chi}}_{/\sim
K},\quad\gamma^{L}\mapsto\gamma^{G},$

is a bijection.

We now use the results of [1], Chapter 16, using what is called there a
translation datum to reduce our main result to the case of no singular
infinitesimal character. The translation datum consists of our singular
infinitesimal character $\xi$, a weight $\mu$ for $H^{\flat}\cap
M^{\flat}=T^{\flat}A_{M^{\flat}}$ satisfying $\xi^{\prime}=\xi+\mu$ such that

  * a)

$\xi^{\prime}$ is non-singular for $\mathfrak{l}^{\flat}$

  * b)

If $\langle\check{\alpha},\xi\rangle$ is a positive integer for $\alpha\in
R(\mathfrak{g},\mathfrak{h}^{\flat})$, then $\langle\check{\alpha},\xi\rangle$
is a positive integer.

Then by ibid, (16.5)(a) and (16.5)(d), there is a injection
$\iota^{L^{\flat}}_{\xi,\xi^{\prime}}:\mathcal{P}^{L^{\flat}}_{\xi}\rightarrow\mathcal{P}^{L^{\flat}}_{\xi^{\prime}}$
respecting $K_{M}$-conjugacy classes such that for
$\gamma^{L^{\flat}},\,\delta^{L^{\flat}}\in\mathcal{P}^{L^{\flat}}_{\xi}$ :

$M^{L^{\flat}}(\gamma^{L^{\flat}},\delta^{L^{\flat}})=M^{L^{\flat}}(\iota^{l^{\flat}}_{\xi,\xi^{\prime}}(\gamma^{L^{\flat}}),\iota^{l^{\flat}}_{\xi,\xi^{\prime}}(\delta^{L^{\flat}})).$

We can use the same translation datum, but this time for $G$, and we get a
injection
$\iota^{G}_{\xi,\xi^{\prime}}:\mathcal{P}^{G}_{\xi}\rightarrow\mathcal{P}^{G}_{\xi^{\prime}}$
respecting $K$-conjugacy classes such that for
$\gamma^{G},\,\delta^{G}\in\mathcal{P}^{G}_{\xi}$ :

$M^{G}(\gamma^{G},\delta^{G})=M^{G}(\iota^{G}_{\xi,\xi^{\prime}}(\gamma^{G}),\iota_{\xi,\xi^{\prime}}(\delta^{G})).$

By the result for non singular infinitesimal character proved in the previous
section, we have that

$M^{L^{\flat}}(\iota^{l^{\flat}}_{\xi,\xi^{\prime}}(\gamma^{L^{\flat}}),\iota^{l^{\flat}}_{\xi,\xi^{\prime}}(\delta^{L^{\flat}}))=M^{G}(\iota^{G}_{\xi,\xi^{\prime}}(\gamma^{G}),\iota_{\xi,\xi^{\prime}}(\delta^{G})).$

Since

$\textstyle{\mathcal{P}^{L^{\flat}}_{\xi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota^{L^{\flat}}_{\xi,\xi^{\prime}}}$$\textstyle{\mathcal{P}^{L^{\flat}}_{\xi^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{P}^{G}_{\xi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota^{G}_{\xi,\xi^{\prime}}}$$\textstyle{\mathcal{P}^{G}_{\xi^{\prime}}}$

is a commutative diagram, where the vertical maps are the one-to-one maps
previously defined, we get

$M^{L^{\flat}}(\gamma^{L^{\flat}},\delta^{L^{\flat}})=M^{G}(\gamma^{G},\delta^{G})$

and Thm. 4.2 is also proved, with the conditions on $\nu$ strengthened by as
above.

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