# Clifford algebras, symmetric spaces and cohomology rings of Grassmannians

Kieran Calvert  Department of Mathematics and Statistics, Fylde Building,
Lancaster University, Lancaster, UK<EMAIL_ADDRESS>,  Kyo
Nishiyama  Department of Mathematics, Aoyama Gakuin University, Fuchinobe
5-10-1, Chuo-ku, Sagamihara 252-5258, Japan<EMAIL_ADDRESS>and  Pavle
Pandžić  Department of Mathematics, Faculty of Science, University of Zagreb,
Bijenička 30, 10000 Zagreb, Croatia<EMAIL_ADDRESS>Dedicated to Bert Kostant

###### Abstract.

We study various kinds of Grassmannians or Lagrangian Grassmannians over
$\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$, all of which can be expressed as
$\mathbb{G}/\mathbb{P}$ where $\mathbb{G}$ is a classical group and
$\mathbb{P}$ is a parabolic subgroup of $\mathbb{G}$ with abelian unipotent
radical. The same Grassmannians can also be realized as (classical) compact
symmetric spaces $G/K$. We give explicit generators and relations for the de
Rham cohomology rings of $\mathbb{G}/\mathbb{P}\cong G/K$. At the same time we
describe certain filtered deformations of these rings, related to Clifford
algebras and spin modules. While the cohomology rings are of our primary
interest, the filtered setting of $K$-invariants in the Clifford algebra
actually provides a more conceptual framework for the results we obtain.

###### Key words and phrases:

Grassmannian, Lagrangian Grassmannian, symmetric space, de Rham cohomology,
spin module, Clifford algebra, exterior algebra, invariants

###### 2020 Mathematics Subject Classification:

primary: 14M15, 57T15, 53C35, 22E47. secondary: 32M15, 15A66

K. Nishiyama is supported by JSPS KAKENHI Grant Number #21K03184.

P. Pandžić is supported by the QuantiXLie Center of Excellence, a project
cofinanced by the Croatian Government and European Union through the European
Regional Development Fund - the Competitiveness and Cohesion Operational
Programme (KK.01.1.1.01.0004).

##  1\. Introduction

This paper is motivated by email correspondence between the third named author
and Bert Kostant in 2004 [Kos04]. In his study of the action of Dirac
operators on Harish-Chandra modules attached to a real reductive Lie group
$G_{\mathbb{R}}$, the third named author was led to consider the algebra
$C(\mathfrak{p})^{K}$. Here $K$ is a maximal compact subgroup of
$G_{\mathbb{R}}$ corresponding to a Cartan decomposition
$\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$ of the complexified Lie algebra
of $G_{\mathbb{R}}$ and $C(\mathfrak{p})$ is the Clifford algebra of
$\mathfrak{p}$ with respect to the (extended) Killing form. The third named
author remembered hearing Kostant speak about the graded version of
$C(\mathfrak{p})^{K}$, $(\textstyle{\bigwedge}\mathfrak{p})^{K}$, and its
relation to cohomology, so he asked Kostant about the latter algebra expecting
the description of the graded version of the algebra will give some
information about the algebra $C(\mathfrak{p})^{K}$ he was interested in.
Kostant replied as follows:

From<EMAIL_ADDRESS>Mon Mar 15 21:07:54 2004

for<EMAIL_ADDRESS>Mon, 15 Mar 2004 21:07:53 +0100 (MET)

Subject: Re: a question

Dear Pavle

The following is known. In general (\wedge p) $\boldsymbol{\land}$ k is the

cochain complex whose cohomology is H(G/K). In the symmetric case (G/K is a
symmetric space) the cobounbary operator is trivial so that (\wedge p)
$\boldsymbol{\land}$ k = H(G/K). If in addition rank K = rank G then all
cohomology is even dimension and if e = Euler characteristic of G/K then of
course
dim (\wedge p) $\boldsymbol{\land}$ k = e. Also in this case dim p is even and
C(p) = End S where S is the spin module. Furthermore under the action of K

S=V_1 + .... + V_e

where the V_i are irreducible K modules and all distinct. Thus C(p)
$\boldsymbol{\land}$ K is an abelian (semisimple) algebra of dimension e. This
is a special case of my result with Sternberg, Ramond and Gross (GKRS) in the
PNAS. It is a nice unsolved problem to locate the 1-dimensional idempotents
which project onto the various V_ i. In case G/K is Hermitian symmetric these
idempotents I believe correspond to the Schubert classes in (\wedge p)
$\boldsymbol{\land}$ k. If so one has some sort of generalization of Schubert
classes when G/K is not Hermitian.

Best regards

Bert

It turns out Kostant was not exactly right in thinking that the idempotents
will correspond to the Schubert classes; in fact, they typically all have
nonzero top degree term, and one must take their linear combinations in order
to get a basis compatible with filtration. His intuition that this question is
related to some sort of generalization of Schubert classes when $G/K$ is not
Hermitian was however right; as we shall see, these cases correspond to
various kinds of real or quaternionic Grassmannians which possess their own
Schubert calculus.

Our main goal in this paper is to describe the de Rham cohomology rings of
these Grassmannians using their realization as compact symmetric spaces. The
main tool is the representations of the Clifford algebras associated with
symmetric spaces. When the Grassmannians are complex, the results we obtain
here are well known. However, for the real and quaternionic Grassmannians, the
results are not widely known. For example, the ring structure of the
cohomology of the real Grassmannians was conjectured by Casian and Kodama
[CK13]. Recently, many related papers have appeared (see [Rab16, LR22, RSM19,
Mat21] for integral cohomology, for example), but these papers treat the
Grassmannians individually, and not in a uniform way. See also [Che20, CHL20,
EH13, HL11] for identifications of compact symmetric spaces and Grassmannians.

We study these cohomology rings systematically. The key ideas are the usage of
Clifford algebras as mentioned above and the description of the Grassmannians
as flag manifolds corresponding to maximal parabolic subgroups with abelian
unipotent radicals. In fact, we start with the Grassmannian
$\mathbb{G}/\mathbb{P}$, where $\mathbb{P}$ is a maximal parabolic subgroup
with abelian unipotent radical in a reductive Lie group $\mathbb{G}$, then
produce symmetric spaces of compact and noncompact type using three
involutions $\theta,\sigma$ and $\tau=\theta\sigma$ of $\mathbb{G}$, which are
mutually commuting.

Our results describe the de Rham cohomology ring of each of the Grassmannians
in our list by explicit generators and relations, and also give an explicit
basis consisting of certain monomials. In most cases (including the well known
complex Grassmannian cases) we show that our basis can be replaced by a basis
consisting of certain Schur polynomials. These have the advantage of a rather
well understood multiplication table, related to the Littlewood-Richardson
coefficients. However our monomials with their clear structure of generators
and relations also lead to an explicit multiplication table, as explained in
Section 4. In this way we get an alternative approach to Schubert calculus on
the Grassmannians in question.

The paper is organized as follows.

In Section 2 we describe our Grassmannians $\mathbb{G}/\mathbb{P}$ and give
their realizations as compact symmetric spaces. The cases are summarized in
the table at the end of the introduction. In each case $\mathbb{P}$ is a
maximal parabolic subgroup of $\mathbb{G}$ with abelian unipotent radical.
Note that a Grassmannian is the set of certain subspaces of fixed dimension in
a vector space $V$. On this set of subspaces, the automorphism group
$\mathbb{G}$ of $V$, which typically preserves additional structure (quadratic
or symplectic forms), acts transitively. This is justified by the following
well known theorem by Witt.

######  Theorem 1.1.

[Wit37], [Bou74, Ch. 1-2], [Die55]. Let $V$ be a vector space over
$\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$ with a nondegenerate form
$\langle\,,\rangle$ which is either bilinear symmetric, or bilinear skew
symmetric, or Hermitian, or skew Hermitian. Let $U$ and $W$ be subspaces of
$V$ and let $\varphi:U\to W$ be an isomorphism preserving $\langle\,,\rangle$.
Then $\varphi$ extends to an automorphism of $V$ preserving
$\langle\,,\rangle$.

Let $\mathbb{P}$ be the stabilizer in $\mathbb{G}$ of a standard subspace $U$
in our Grassmannian. (In some cases, $U$ is a Lagrangian or isotropic
subspace. It depends on the situation.) Then $\mathbb{P}$ is a maximal
parabolic subgroup and our Grassmannian is equal to $\mathbb{G}/\mathbb{P}$.
So Grassmannians are naturally identified with (partial) flag manifolds
$\mathbb{G}/\mathbb{P}$.

As already mentioned, compact symmetric spaces are diffeomorphic to
Grassmannians $\mathbb{G}/\mathbb{P}$ where $\mathbb{P}$ has abelian unipotent
radical. In fact, these Grassmannians exhaust all such pairs
$(\mathbb{G},\mathbb{P})$ (see [Wol76], [How95, § 5.5.1], and also [RRS92]).
Then, in appropriate realizations, the Grassmannians are varieties of either
ordinary subspaces of a vector space, or of Lagrangian subspaces with respect
to a certain form.

In this paper, the group $\mathbb{G}$ will be a classical group; in
particular, it is a reductive matrix group, and we always consider the
standard Cartan involution

(1.2) $\theta(g)=(\bar{g}^{t})^{-1},\quad g\in\mathbb{G}.$

The corresponding maximal compact subgroup $\mathbb{G}^{\theta}$ will be
denoted by $\mathbb{K}$ and also by $G$.

We will use Proposition 2.4 below to see that $\mathbb{K}$ acts transitively
on the Grassmannian, so that $\mathbb{G}/\mathbb{P}$ is diffeomorphic to
$\mathbb{K}/\mathbb{P}\cap\mathbb{K}$. This follows easily from the fact that
$\mathbb{P}$ contains a minimal parabolic subgroup
$\mathbb{P}_{0}=\mathbb{M}\mathbb{A}\mathbb{N}_{0}$, and that $\mathbb{G}$ has
an Iwasawa decomposition $\mathbb{G}=\mathbb{K}\mathbb{A}\mathbb{N}_{0}$,
where $\mathbb{K}$ is as above.

It is known by [TK68, §4] (see also [Kob08, Lemma 7.3.1] and [RRS92]) that the
following are equivalent:

  1. (a)

$\mathbb{P}$ has abelian unipotent radical;

  2. (b)

$\mathbb{P}$ has a Levi subgroup $\mathbb{L}$ which is a symmetric subgroup of
$\mathbb{G}$;

  3. (c)

$K:=\mathbb{P}\cap\mathbb{K}=\mathbb{L}\cap\mathbb{K}$ is a symmetric subgroup
of $G=\mathbb{K}$.

We give a short and comprehensive proof of this result in Theorem 2.2 for
convenience of the readers.

The involution $\sigma$ of $\mathbb{G}$ mentioned above is related to the
above Levi subgroup $\mathbb{L}$ of $\mathbb{P}$: $\mathbb{L}$ is
$\mathbb{G}^{\sigma}$, the subgroup of $\mathbb{G}$ consisting of points fixed
by $\sigma$; we will see that also $\mathbb{P}$ itself is $\sigma$-stable. The
involution $\sigma$, which we describe explicitly below, commutes with the
Cartan involution $\theta$, and hence
$K=\mathbb{L}^{\theta}=\mathbb{L}\cap\mathbb{K}$ is a maximal compact subgroup
of $\mathbb{L}$.

Let us denote by $\tau=\theta\sigma$ the third involution of $\mathbb{G}$. We
denote the fixed point subgroup $\mathbb{G}^{\tau}$ by $G_{\mathbb{R}}$, so
that $G_{\mathbb{R}}/K$ is a noncompact Riemannian symmetric space with its
compact dual equal to $G/K$. It turns out that in this way we get to cover the
full list of compact classical symmetric spaces as listed e.g. in [How95,
p.69].

We summarize the three involutions and symmetric spaces thus obtained in the
following diagram. See also Table 1 at the end of Introduction.

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\mathbb{P}=\mathbb{L}\mathbb{N}:\ &\text{$\sigma$-stable parabolic
subgroup}\\\ &\text{with abelian unipotent radical}\\\ \tau=\theta&\text{ on
}\mathbb{L}\\\ \theta=\sigma&\text{ on }G_{\mathbb{R}}\\\ \sigma=\tau&\text{
on }\mathbb{K}=G\end{aligned}$

Since $\mathbb{P}$ is block upper triangular with two diagonal blocks, of
sizes (say) $p$ and $q$, $\mathbb{L}$ can be taken as the block diagonal part
of $\mathbb{P}$. Now if we denote by $I_{p,q}$ the matrix
$\left(\begin{smallmatrix}I_{p}&0\cr 0&-I_{q}\end{smallmatrix}\right)$, then

$I_{p,q}\begin{pmatrix}A&B\cr
C&D\end{pmatrix}I_{p,q}=\begin{pmatrix}A&-B\cr-C&D\end{pmatrix},$

and we see that the involution we want is

(1.3) $\sigma(g)=I_{p,q}gI_{p,q}.$

It follows that $\mathbb{P}$ is $\sigma$-stable, and it will now be very easy
to identify the groups $\mathbb{L}=\mathbb{G}^{\sigma}$. (In fact, we will see
in Section 2 that $\sigma$ can be described in terms of $\mathbb{P}$ only; see
Theorem 2.2.)

It turns out that the complexifications of these groups are exactly the groups
listed in [How95, p. 70] as the groups acting in a skew-multiplicity free way
on $\mathfrak{p}$, the complexified tangent space of
$G/K\simeq\mathbb{G}/\mathbb{P}$ at the base point $eK$. Here “skew-
multiplicity free” means that $\mathbb{L}_{\mathbb{C}}$ acts on
$\textstyle{\bigwedge}\mathfrak{p}$ in a multiplicity free way (note that
$\mathfrak{p}$ can also be identified with the complexification of the Lie
algebra of $\mathbb{N}$, so that $\mathbb{L}$ acts on it naturally).

We will also describe the groups $G_{\mathbb{R}}=\mathbb{G}^{\tau}$, where
$\tau={\sigma\theta}$. Note that by (1.2) and (1.3), we get

(1.4) $\sigma\theta(g)=I_{p,q}(\bar{g}^{t})^{-1}I_{p,q}.$

The group $G_{\mathbb{R}}$ is a noncompact reductive Lie group with maximal
compact subgroup $K=\mathbb{K}^{\sigma}$, and
$G/K=\mathbb{K}/\mathbb{K}^{\sigma}$ is the compact dual of the noncompact
Riemannian symmetric space $G_{\mathbb{R}}/K$ as explained above. In this way,
we get to cover the full list of noncompact classical symmetric spaces. As in
the Hermitian symmetric case, the noncompact Riemannian symmetric space
$G_{\mathbb{R}}/K$ is embedded into $\mathbb{G}/\mathbb{P}$ as an open subset
(a generalization of the Borel embedding).

The realizations of our Grassmannians as symmetric spaces are known in most
(or all) cases, but the results are scattered in the literature. We will
indicate some references when we get to the case by case analysis. Our view
point is to produce the classical Riemannian symmetric spaces, both compact
and noncompact ones, in terms of the pairs $(\mathbb{G},\mathbb{P})$ on our
list.

In Section 3 we collect some facts needed for our description of the
cohomology rings of compact symmetric spaces $G/K$ as above (and thus also of
the corresponding Grassmannians $\mathbb{G}/\mathbb{P}$). For $\sigma$ as
above, its restriction to $G$ is an involution such that $K=G^{\sigma}$. Let
$\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$ be the decomposition of the
complexified Lie algebra $\mathfrak{g}$ of $G$ into eigenspaces of $\sigma$.
We are assuming $G$ is connected, but $K$ need not be connected.

As noted in Kostant’s email message, if $\mathfrak{g}$ and $\mathfrak{k}$ have
equal rank, then the algebras $C(\mathfrak{p})^{K}$ and
$(\textstyle{\bigwedge}\mathfrak{p})^{K}$ can be expressed as

$C(\mathfrak{p})^{K}=\operatorname{Pr}(S);\qquad(\textstyle{\bigwedge}\mathfrak{p})^{K}=\operatorname{gr}\operatorname{Pr}(S),$

where the algebra $\operatorname{Pr}(S)$ is spanned by the projections of the
spin module $S$ to its isotypic components for the pin double cover
$\widetilde{K}$ of $K$. As explained in Subsections 3.1 and 3.3, one can use
the natural map $\alpha:U(\mathfrak{k})\to C(\mathfrak{p})$ that gives the
spin module its $\mathfrak{k}$-module structure, and the fact that the
projections are given by the action of the center of $U(\mathfrak{k})$, to
express the algebra $\operatorname{Pr}(S)$ as the quotient of
$\mathbb{C}[\mathfrak{t}^{*}]^{W_{K}}$ by the ideal generated by the elements
of $\mathbb{C}[\mathfrak{t}^{*}]^{W_{G}}$ vanishing at $\rho$. Here
$\mathfrak{t}$ is the complexification of a Cartan subalgebra of the Lie
algebra of $K$, $W_{K}$ is the Weyl group of $K$ (see (3.30)),
$W_{G}=W_{\mathfrak{g}}$ is the Weyl group of $G$ or equivalently the Weyl
group of the root system $\Delta(\mathfrak{g},\mathfrak{t})$, and $\rho$ is
the half sum of roots in the (fixed) positive root system
$\Delta^{+}(\mathfrak{g},\mathfrak{t})$. The algebra
$\operatorname{gr}\operatorname{Pr}(S)$ attached to the natural filtration of
$\operatorname{Pr}(S)$ can be expressed as the quotient of
$\mathbb{C}[\mathfrak{t}^{*}]^{W_{K}}$ by the ideal generated by the elements
of $\mathbb{C}[\mathfrak{t}^{*}]^{W_{G}}$ vanishing at $0$.

In the “almost equal rank case”
$(\mathfrak{g},\mathfrak{k})=(\mathfrak{so}(2p+2q+2),\mathfrak{so}(2p+1)\times\mathfrak{so}(2q+1))$,
the algebras $C(\mathfrak{p})^{K}$ and
$(\textstyle{\bigwedge}\mathfrak{p})^{K}$ are very close to
$\operatorname{Pr}(S)$ and $\operatorname{gr}\operatorname{Pr}(S)$; one has to
tensor with the Clifford respectively exterior algebra of a certain one-
dimensional space. This is explained in Subsections 3.5 and 3.7.

On the opposite end are the primary and almost primary cases; in these cases
the $\widetilde{K}$-module $S$ has only one isotypic component and the algebra
of projections is trivial. These cases are described in Subsections 3.6 and
3.8. The algebra $(\textstyle{\bigwedge}\mathfrak{p})^{K}$ is now equal to the
exterior algebra of a certain subspace of
$(\textstyle{\bigwedge}{\mathfrak{p}})^{K}$ (denoted by $\cal
P_{\wedge}(\mathfrak{p})$), where $\cal P_{\wedge}(\mathfrak{p})$ is the
subspace orthogonal to the square of the augmentation ideal (Definition 3.26).
This result is due to Hopf and Samelson [Hop41, Sam41] for the group case and
Theorem 3.29 for the remaining (almost) primary cases. Similar isomorphisms go
way back to Cartan, Chevalley, Koszul and others; see [GHV76, p. 568]. We
believe that likewise $C(\mathfrak{p})^{K}$ is the Clifford algebra over $\cal
P_{\wedge}(\mathfrak{p})$, but this is currently known (by the results of
Kostant [Kos97]) only in the group cases, i.e., when
$\mathfrak{g}=\mathfrak{g}_{1}\oplus\mathfrak{g}_{1}$ and
$\mathfrak{k}\cong\mathfrak{p}\cong\mathfrak{g}_{1}$.

Finally, in Section 4 we give a precise description, by generators and
relations, of the algebras $(\textstyle{\bigwedge}\mathfrak{p})^{K}$ in each
of the cases. In the equal rank and almost equal rank cases, we also describe
the algebras $C(\mathfrak{p})^{K}$, which in these cases amounts to describing
the algebras $\operatorname{Pr}(S)$. We also give explicit bases for these
algebras. In this way we get to compute the de Rham cohomology of the
symmetric spaces on our list, and thus also of the corresponding
Grassmannians.

Namely, as mentioned in Kostant’s email message, the cohomology of the compact
symmetric space $G/K$ is (after complexification) equal to
$(\textstyle{\bigwedge}\mathfrak{p})^{K}$. This fact is quite well known, but
it is not easy to find an appropriate reference. It is proved in [Tay18]
(unpublished) using Hodge theory, partially proved in [Leu16], and proved in
[GHV76] under the assumption $K$ is connected. It is also mentioned in passing
in [How95, p. 69], and in [BW00, § 1.6]. Borel and Wallach [BW00] attribute
the result to É. Cartan and de Rham. We start Section 4 by presenting a simple
proof of this fact which we learned from Sebastian Goette [Goe23].

We are thus led to study the appropriate quotients of the $W_{K}$-invariants
in $\mathbb{C}[\mathfrak{t}^{*}]$ in each of the cases. The results often
involve the following algebra.

######  Definition 1.5.

Let $p,q\in\mathbb{Z}$ with $1\leq p\leq q$, and let
$c=(c_{1},\dots,c_{p+q})\in\mathbb{C}^{p+q}$. We define $\mathfrak{H}(p,q;c)$
to be the algebra generated by $r_{1},\dots,r_{p}$ and $s_{1},\dots,s_{q}$
with relations generated by

$\sum_{i,j\geq 0;\ i+j=k}r_{i}s_{j}=c_{k}$

for $k=1,\dots,p+q$, where we set $r_{0}=s_{0}=1$ and $r_{i}=0$ if $i>p$,
$s_{j}=0$ if $j>q$.

We can use the first $q$ of the relations to express $s_{1},\dots,s_{q}$ in
terms of the $r_{i}$, so $\mathfrak{H}(p,q;c)$ is in fact generated by
$r_{1},\dots,r_{p}$ only. The remaining relations can be used to obtain the
relations among the $r_{i}$, not involving the $s_{j}$. We do that in the
proof of Theorem 4.5; the relations among the $r_{i}$ are (4.9) and (4.14) and
they form another set of defining relations. From these relations one can
obtain expressions for each monomial in $r_{1},\dots,r_{p}$ of degree $q+1$ as
a linear combination of lower degree monomials in $r_{1},\dots,r_{p}$. We will
also see that the monomials in the $r_{i}$ of degree at most $q$ form a basis
of the algebra $\mathfrak{H}(p,q;c)$; so $\mathfrak{H}(p,q;c)$ can be
identified with the space

$\mathbb{C}[r_{1},\dots,r_{p}]_{\leq q}$

of polynomials in the $r_{i}$ of degree $\leq q$.

We show in Remark 4.16 that the above monomials span the same subspace of
$\mathbb{C}[r_{1},\dots,r_{p}]$ as the Schur polynomials $s_{\lambda}$
attached to partitions $\lambda$ with Young diagrams contained in the $p\times
q$ box. Moreover, our basis consisting of monomials and the basis consisting
of Schur polynomials are connected by a triangular change of basis. In this
way we get a connection with the usual Schubert calculus, where the
multiplication of the Schur polynomials is given in terms of Littlewood-
Richardson coefficients.

For $G/K=\operatorname{U}(p+q)/\operatorname{U}(p)\times\operatorname{U}(q)$,
we prove in Theorem 4.5 that $C(\mathfrak{p})^{K}$ is isomorphic to the
algebra $\mathfrak{H}(p,q;c)$, with the $r_{i}$ being the elementary symmetric
functions in the first $p$ coordinate functions $x_{1},\dots,x_{p}$ on
$\mathfrak{t}\cong\mathbb{C}^{p+q}$, the $s_{j}$ being the elementary
symmetric functions in the last $q$ coordinate functions
$x_{p+1},\dots,x_{p+q}$ on $\mathfrak{t}$, and
$c=(t_{1}(\rho),\dots,t_{p+q}(\rho))$, where $t_{k}$ are the elementary
symmetric functions on $x_{1},\dots,x_{p+q}$. The natural filtration on
$C(\mathfrak{p})^{K}$ coming from the filtration of $C(\mathfrak{p})$
corresponds to the filtration on $\mathfrak{H}(p,q;c)$ obtained by setting

(1.6) $\deg r_{i}=2i,\qquad i=1,\dots,p.$

The cohomology ring $(\textstyle{\bigwedge}\mathfrak{p})^{K}$ is isomorphic to
$\mathfrak{h}(p,q,0)$, and its natural grading is again obtained by (1.6).

For
$G/K=\operatorname{Sp}(p+q)/\operatorname{Sp}(p)\times\operatorname{Sp}(q)$
(Theorem 4.19), the algebra $C(\mathfrak{p})^{K}$ is again
$\mathfrak{H}(p,q;c)$, but now the $r_{i}$, the $s_{j}$ and the $t_{k}$ are
elementary symmetric functions on the squares of the appropriate variables.
The parameter $c$ is again given by evaluating $t_{k}$ at $\rho$. The
filtration is now obtained by setting $\deg r_{i}=4i$, $i=1,\dots,p$. The
cohomology ring $(\textstyle{\bigwedge}\mathfrak{p})^{K}$ is again isomorphic
to $\mathfrak{h}(p,q,0)$, and its natural grading is also obtained by setting
$\deg r_{i}=4i$. For
$G/K=\operatorname{SO}(k+m)/S(\operatorname{O}(k)\times\operatorname{O}(m))$
(Theorem 4.21), the algebra $C(\mathfrak{p})^{K}$ is $\mathfrak{H}(p,q;c)$ if
$(k,m)=(2p,2q)$ or $(2p,2q+1)$, with $\\{r_{i}\\}$, $\\{s_{j}\\}$,
$\\{t_{k}\\}$ and $c$ defined similarly as above. If $(k,m)=(2p+1,2q+1)$ (the
almost equal rank case), there is an extra generator $e$, of degree $2p+2q+1$,
squaring to $1$. The filtration degrees of the $r_{i}$ are again equal to
$4i$. The cohomology ring $(\textstyle{\bigwedge}\mathfrak{p})^{K}$ is
isomorphic to $\mathfrak{H}(p,q,0)$ or
$\mathfrak{H}(p,q,0)\oplus\mathfrak{H}(p,q;0)e$, and its natural grading is
also obtained by setting $\deg r_{i}=4i$ and $\deg e=2p+2q+1$. In this case we
get to prove the conjecture of Casian-Kodama [CK13].

For $G/K=\operatorname{U}(n)/\operatorname{O}(n)$ (Theorem 3.29), the
situation is different: the algebra $(\textstyle{\bigwedge}\mathfrak{p})^{K}$
is the exterior algebra on the subspace $\cal P_{\wedge}(\mathfrak{p})$
(Definition 3.26), and the degrees are given in Table 2.

For $G/K=\operatorname{Sp}(n)/U(n)$ (Theorem 4.25), we are back to elementary
symmetric functions: $r_{1},\dots,r_{n}$ are the elementary symmetric
functions on the coordinate functions $x_{1},\dots,x_{n}$ on the Cartan
subalgebra $\mathfrak{t}\cong\mathbb{C}^{n}$ of $\mathfrak{k}$, while
$t_{1},\dots,t_{n}$ are the elementary symmetric functions on the squares of
the $x_{i}$. The algebras $C(\mathfrak{p})^{K}$ and
$(\textstyle{\bigwedge}\mathfrak{p})^{K}$ are generated by the $r_{i}$, but
the relations are now different:

$r_{k}^{2}=t_{k}+2r_{k-1}r_{k+1}-2r_{k-2}r_{k+2}+\dots,\quad k=1,\dots,n,$

where as usual we set $r_{0}=1$ and $r_{i}=0$ for $i>n$ or $i<0$, and where
$t_{k}$ should be replaced by $t_{k}(\rho)$ if the algebra is
$C(\mathfrak{p})^{K}$ and with 0 if the algebra is
$(\textstyle{\bigwedge}\mathfrak{p})^{K}$. This time a basis for each of our
algebras is given by the monomials

$r_{1}^{{\varepsilon}_{1}}r_{2}^{{\varepsilon}_{2}}\dots
r_{n}^{{\varepsilon}_{n}},\quad{\varepsilon}_{i}\in\\{0,1\\}.$

The filtration degree of $C(\mathfrak{p})^{K}$ inherited from
$C(\mathfrak{p})$, and the gradation degree of
$(\textstyle{\bigwedge}\mathfrak{p})^{K}$ inherited from
$\textstyle{\bigwedge}\mathfrak{p}$, are obtained by setting $\deg r_{i}=2i$
for $i=1,\dots,n$.

For $G/K=\operatorname{SO}(2n)/\operatorname{U}(n)$ (Theorem 4.28) the
situation is entirely analogous to the case $G/K=\operatorname{Sp}(n)/U(n)$,
except that we get to eliminate $r_{n}$ from the list of generators.

For the group cases $G\times G/\Delta G\cong G$ where $G$ is
$\operatorname{SO}(n)$, $\operatorname{U}(n)$ or $\operatorname{Sp}(n)$
(Theorem 4.30), the algebras $C(\mathfrak{p})^{K}\cong
C(\mathfrak{g})^{\mathfrak{g}}$ and
$(\textstyle{\bigwedge}\mathfrak{p})^{K}\cong(\textstyle{\bigwedge}\mathfrak{g})^{\mathfrak{g}}$
are the Clifford respectively exterior algebra of the graded subspace $\cal
P_{\wedge}(\mathfrak{p})$ (Definition 3.26).

For the Clifford algebras, these cases were settled by Kostant in [Kos97]. For
the exterior algebras, the fact that
$(\textstyle{\bigwedge}{\mathfrak{p}})^{\mathfrak{k}}$ is isomorphic to a
graded subspace goes back to Cartan, Chevalley, Koszul and others [GHV76]. The
degrees are given in Table 2.

For the cases $G/K=\operatorname{U}(2n)/\operatorname{Sp}(n)$ (Theorem 4.31)
the algebra $(\textstyle{\bigwedge}\mathfrak{p})^{K}$ is the exterior algebra
of the graded subspace $\cal P_{\wedge}(\mathfrak{p})$ (Definition 3.26).
Again, the degrees are given in Table 2.

Our results are well known for complex Grassmannians, i.e., for
$\operatorname{Gr}_{p}(\mathbb{C}^{p+q})\cong\operatorname{U}(p+q)/\operatorname{U}(p)\times\operatorname{U}(q)$,
$\operatorname{Gr}_{2}(\mathbb{R}^{2+q})\cong\operatorname{SO}(2+q)/S(\operatorname{O}(2)\times\operatorname{O}(q))$,
$\operatorname{LGr}(\mathbb{C}^{2n})\cong\operatorname{Sp}(n)/\operatorname{U}(n)$
and
$\operatorname{OLGr}^{+}(\mathbb{C}^{2n})\cong\operatorname{SO}(2n)/\operatorname{U}(n)$.
Among many papers dealing with the complex Grassmannians and their Schubert
calculus, we mention [Ful97], [FP98], [Tam05, Tam01], and [Pra91, PR96, PR03].

Table 1. Table of Grassmannians and corresponding symmetric spaces.

General Linear $n=p+q$ |  |  |  |  |  |
---|---|---|---|---|---|---
$\mathbb{G}$ | $\mathbb{P}$ | $\mathbb{L}=\mathbb{G}^{\sigma}$ | $\mathbb{K}=\mathbb{G}^{\theta}=G$ | $\mathbb{G}^{\sigma\theta}=G_{\mathbb{R}}$ | $\mathbb{K}^{\sigma}=G_{\mathbb{R}}^{\theta}=K$ | $\mathfrak{p}_{0}$
$\operatorname{GL}_{n}(\mathbb{R})$ | $\mathrm{Stab}(\mathbb{R}^{p})$ | $\operatorname{GL}_{p}(\mathbb{R}){\times}\operatorname{GL}_{q}(\mathbb{R})$ | $\operatorname{U}_{n}(\mathbb{R})$ | $\operatorname{U}_{p,q}(\mathbb{R})$ | $\operatorname{U}_{p}(\mathbb{R}){\times}\operatorname{U}_{q}(\mathbb{R})$ | $\operatorname{Mat}_{p,q}(\mathbb{R})$
$\operatorname{GL}_{n}(\mathbb{C})$ | $\mathrm{Stab}(\mathbb{C}^{p})$ | $\operatorname{GL}_{p}(\mathbb{C}){\times}\operatorname{GL}_{q}(\mathbb{C})$ | $\operatorname{U}_{n}(\mathbb{C})$ | $\operatorname{U}_{p,q}(\mathbb{C})$ | $\operatorname{U}_{p}(\mathbb{C}){\times}\operatorname{U}_{q}(\mathbb{C})$ | $\operatorname{Mat}_{p,q}(\mathbb{C})$
${\operatorname{GL}}_{n}(\mathbb{H})$ | $\mathrm{Stab}(\mathbb{H}^{p})$ | $\operatorname{GL}_{p}(\mathbb{H}){\times}\operatorname{GL}_{q}(\mathbb{H})$ | $\operatorname{U}_{n}(\mathbb{H})$ | $\operatorname{U}_{p,q}(\mathbb{H})$ | $\operatorname{U}_{p}(\mathbb{H}){\times}\operatorname{U}_{q}(\mathbb{H})$ | $\operatorname{Mat}_{p,q}(\mathbb{H})$
Symplectic |  |  |  |  |  |
$\mathbb{G}$ | $\mathbb{P}$ | $\mathbb{L}=\mathbb{G}^{\sigma}$ | $\mathbb{K}=\mathbb{G}^{\theta}=G$ | $\mathbb{G}^{\sigma\theta}=G_{\mathbb{R}}$ | $\mathbb{K}^{\sigma}=G_{\mathbb{R}}^{\theta}=K$ | $\mathfrak{p}_{0}$
$\operatorname{Sp}_{2n}(\mathbb{R})$ | $\mathrm{Stab}(L_{0})$ | $\operatorname{GL}_{n}(\mathbb{R})$ | $\operatorname{U}_{n}(\mathbb{C})$ | $\operatorname{GL}_{n}(\mathbb{R})$ | $\operatorname{U}_{n}(\mathbb{R})$ | $\operatorname{Sym}_{n}(\mathbb{R})$
$\operatorname{Sp}_{2n}(\mathbb{C})$ | $\mathrm{Stab}(L_{0})$ | $\operatorname{GL}_{n}(\mathbb{C})$ | $\operatorname{U}_{n}(\mathbb{H})$ | $\operatorname{Sp}_{2n}(\mathbb{R})$ | $\operatorname{U}_{n}(\mathbb{C})$ | $\operatorname{Sym}_{n}(\mathbb{C})$
Orthogonal |  |  |  |  |  |
$\mathbb{G}$ | $\mathbb{P}$ | $\mathbb{L}=\mathbb{G}^{\sigma}$ | $\mathbb{K}=\mathbb{G}^{\theta}=G$ | $\mathbb{G}^{\sigma\theta}=G_{\mathbb{R}}$ | $\mathbb{K}^{\sigma}=G_{\mathbb{R}}^{\theta}=K$ | $\mathfrak{p}_{0}$
$\operatorname{O}_{2n}(\mathbb{C})$ | $\mathrm{Stab}(L_{0})$ | $\operatorname{GL}_{n}(\mathbb{C})$ | $\operatorname{O}_{2n}(\mathbb{R})$ | $\operatorname{SO}^{*}(2n,j\operatorname{id}_{n})$ | $\operatorname{U}_{n}(\mathbb{C})$ | $\operatorname{Alt}_{n}(\mathbb{C})$
Hermitian |  |  |  |  |  |
$\mathbb{G}$ | $\mathbb{P}$ | $\mathbb{L}=\mathbb{G}^{\sigma}$ | $\mathbb{K}=\mathbb{G}^{\theta}=G$ | $\mathbb{G}^{\sigma\theta}=G_{\mathbb{R}}$ | $\mathbb{K}^{\sigma}=G_{\mathbb{R}}^{\theta}=K$ | $\mathfrak{p}_{0}$
$\operatorname{U}_{n,n}(\mathbb{R})$ | $\mathrm{Stab}(L_{0})$ | $\operatorname{GL}_{n}(\mathbb{R})$ | $\operatorname{U}_{n}(\mathbb{R})^{2}$ | $\operatorname{O}_{n}(\mathbb{C})$ | $\Delta(\operatorname{U}_{n}(\mathbb{R}))$ | $\mathrm{SHer}_{n}(\mathbb{R})$
$\operatorname{U}_{n,n}(\mathbb{C})$ | $\mathrm{Stab}(L_{0})$ | $\operatorname{GL}_{n}(\mathbb{C})$ | $\operatorname{U}_{n}(\mathbb{C})^{2}$ | $\operatorname{GL}_{n}(\mathbb{C})$ | $\Delta(\operatorname{U}_{n}(\mathbb{C}))$ | $\mathrm{SHer}_{n}(\mathbb{C})$
$\operatorname{U}_{n,n}(\mathbb{H})$ | $\mathrm{Stab}(L_{0})$ | $\operatorname{GL}_{n}(\mathbb{H})$ | $\operatorname{U}_{n}(\mathbb{H})^{2}$ | $\operatorname{Sp}_{2n}(\mathbb{C})$ | $\Delta(\operatorname{U}_{n}(\mathbb{H}))$ | $\mathrm{SHer}_{n}(\mathbb{H})$
Skew Hermitian |  |  |  |  |  |
$\mathbb{G}$ | $\mathbb{P}$ | $\mathbb{L}=\mathbb{G}^{\sigma}$ | $\mathbb{K}=\mathbb{G}^{\theta}=G$ | $\mathbb{G}^{\sigma\theta}=G_{\mathbb{R}}$ | $\mathbb{K}^{\sigma}=G_{\mathbb{R}}^{\theta}=K$ | $\mathfrak{p}_{0}$
$\operatorname{SO}^{*}(4n)$ | $\mathrm{Stab}(L_{0})$ | $\operatorname{GL}_{n}(\mathbb{H})$ | $\operatorname{U}_{2n}(\mathbb{C})$ | $\operatorname{GL}_{n}(\mathbb{H})$ | $\operatorname{U}_{n}(\mathbb{H})$ | $\operatorname{Her}_{n}(\mathbb{H})$
Quadric |  |  |  |  |  |
$\mathbb{G}$ | $\mathbb{P}$ | $\mathbb{L}=\mathbb{G}^{\sigma}$ | $\mathbb{K}=\mathbb{G}^{\theta}=G$ | $\mathbb{G}^{\sigma\theta}=G_{\mathbb{R}}$ | $\mathbb{K}^{\sigma}=G_{\mathbb{R}}^{\theta}=K$ | $\mathfrak{p}_{0}$
$\operatorname{SO}_{p+1,q+1}(\mathbb{R})_{e}$ | $P_{1}(Q_{n}(\mathbb{R}))$ | $\operatorname{SO}_{1,1}(\mathbb{R}){\times}\operatorname{SO}_{p,q}(\mathbb{R})$ | $\operatorname{SO}_{p+1}(\mathbb{R}){\times}\operatorname{SO}_{q+1}(\mathbb{R})$ | $\operatorname{SO}_{1,p}(\mathbb{R}){\times}\operatorname{SO}_{q,1}(\mathbb{R})$ | $S(\operatorname{O}_{p}(\mathbb{R}){\times}\operatorname{O}_{q}(\mathbb{R}))$ | $\mathbb{R}^{p-1}\oplus\mathbb{R}^{q-1}$
$\operatorname{SO}_{n+2}(\mathbb{C})$ | $P_{1}(Q_{n}(\mathbb{C}))$ | $S(\operatorname{O}_{2}(\mathbb{C}){\times}\operatorname{O}_{n}(\mathbb{C}))$ | $\operatorname{SO}_{n+2}(\mathbb{R})$ | $\operatorname{SO}_{2,n}(\mathbb{R})$ | $S(\operatorname{O}_{2}(\mathbb{R}){\times}\operatorname{O}_{n}(\mathbb{R}))$ | $\mathbb{C}^{n}=\operatorname{Mat}_{2,n}(\mathbb{R})$

$\begin{array}[]{l l}&Key:\quad n=p+q,\\\ &\mathbb{P}:\text{ maximal parabolic
subgroup with abelian nilpotent
radical},\mathbb{P}=\mathbb{L}\ltimes\mathbb{N}:\text{Levi decomposition},\\\
&\mathbb{L}=\mathbb{G}^{\sigma}:\text{ Levi of }\mathbb{P},\quad
G=\mathbb{K}=\mathbb{G}^{\theta}\subset\mathbb{G}:\text{ maximal
compact},\quad G_{\mathbb{R}}=\mathbb{G}^{\sigma\theta}:\text{ noncompact real
group},\par\\\ &K=G_{\mathbb{R}}^{\theta}=\mathbb{G}^{\sigma,\theta}\subset
G_{\mathbb{R}}:\text{ maximal compact
subgroup,}\quad\mathrm{Lie}(\mathbb{N})\simeq\mathfrak{p}_{0}=\mathrm{Lie}(G)^{-\sigma},\\\
&K=\mathbb{K}\cap\mathbb{P}\>(\text{except for
}\mathbb{G}=\operatorname{SO}_{n+2}(\mathbb{C})\text{ when
}K_{e}=\mathbb{K}\cap\mathbb{P}\text{ and }\mathbb{G}/\mathbb{P}\simeq
G/K_{e}),\\\ &\text{Compact symmetric space:
}G/K=\mathbb{K}/K\simeq\mathbb{G}/\mathbb{P}\text{
Grassmannian},\par\par\par\par\par\\\ &L_{0}:\text{ maximal Lagrangian
subspace },\quad P_{1}(Q_{n}(\mathbb{F})):\text{ stabiliser of a point in
quadric},\par\\\ &\mathrm{Her}_{n}(\mathbb{F}):\text{ Hermitian
matrices},\quad\mathrm{SHer}_{n}(\mathbb{F}):\text{ Skew Hermitian
matrices},\par\\\
&\operatorname{U}_{p,q}(\mathbb{F})=\operatorname{O}({p,q})\text{ if
}\mathbb{F}=\mathbb{R},\>\>\operatorname{U}({p,q})\text{ if
}\mathbb{F}=\mathbb{C},\>\>\operatorname{Sp}({p,q})\text{ if
}\mathbb{F}=\mathbb{H}.\par\end{array}$

##  2\. Realization of certain Grassmannians as compact symmetric spaces

###  2.1. Some general facts

Let $\mathbb{G}$ be one of the groups in Table 1; note that the corresponding
symmetric spaces $G/K$ and $G_{\mathbb{R}}/K$ exhaust the list of classical
symmetric spaces given in [Hel79, Ch.9, Sec.4]. Let $\mathbb{P}$ be the
parabolic subgroup of $\mathbb{G}$ described in Table 1. Then $\mathbb{P}$ has
a Levi decomposition $\mathbb{P}=\mathbb{L}\mathbb{N}$ specified in Table 1.
(As we shall see in the case by case analysis in the subsequent subsections,
$\mathbb{P}$ consists of the block upper triangular matrices in $\mathbb{G}$
with two diagonal blocks, while $\mathbb{L}$ consists of the block diagonal
matrices in $\mathbb{P}$.) Let $\mathfrak{P}=\mathfrak{L}\oplus\mathfrak{N}$
be the corresponding decomposition of the Lie algebra of $\mathbb{P}$. The
opposite parabolic subalgebra is
$\mathfrak{P}^{-}=\mathfrak{L}\oplus\mathfrak{N}^{-}=\mathfrak{L}\oplus\theta\mathfrak{N}$,
where the differential of $\theta$ is still denoted by $\theta$.

The parabolic subgroup $\mathbb{P}$ contains a minimal parabolic subgroup
$\mathbb{P}_{0}=\mathbb{M}\mathbb{A}\mathbb{N}_{0}$ corresponding to an
Iwasawa decomposition $\mathbb{G}=\mathbb{K}\mathbb{A}\mathbb{N}_{0}$. The
Levi subgroup $\mathbb{L}$ of $\mathbb{P}$ contains $\mathbb{M}\mathbb{A}$,
while the unipotent radical $\mathbb{N}$ of $\mathbb{P}$ is contained in
$\mathbb{N}_{0}$. Let $\mathfrak{G},\mathfrak{A},\mathfrak{M}$ be the Lie
algebras of $\mathbb{G},\mathbb{A},\mathbb{M}$. Recall that $\mathfrak{P}$ can
be constructed by taking a subset of simple $(\mathfrak{G},\mathfrak{A})$
roots. Then one generates a root subsystem by these simple roots, which
defines $\mathfrak{L}$ as the span of $\mathfrak{M}\oplus\mathfrak{A}$ and the
root spaces for the roots in this subsystem, and defines $\mathfrak{N}$ to be
the span of the root spaces for the remaining positive roots.

######  Lemma 2.1.

(1) With the above notation, suppose that $\gamma,\delta$ are roots of
$\mathfrak{N}$ such that $\gamma+\delta$ is a root (hence a root of
$\mathfrak{N}$). Then $[\mathfrak{G}_{\gamma},\mathfrak{G}_{\delta}]\neq 0$.

(2) Suppose $\mathfrak{N}$ is abelian. Then
$[\mathfrak{N},\mathfrak{N}^{-}]=[\mathfrak{N},\theta\mathfrak{N}]$ is
contained in $\mathfrak{L}$.

###### Proof.

(1) Let $\delta-k\gamma,\dots,\delta,\delta+\gamma,\dots,\delta+n\gamma$ be
the $\gamma$-string of roots through $\delta$, with $k\geq 0$ and $n\geq 1$.
Let $e\in\mathfrak{G}_{\gamma}$ be nonzero. By [Kna96, Proposition 6.52] there
is an $\mathfrak{sl}_{2}$-triple $e,h,f$ with $h\in\mathfrak{A}$ and
$f\in\mathfrak{G}_{-\gamma}$. Now
$\mathfrak{G}_{\delta-k\gamma}\oplus\dots\oplus\mathfrak{G}_{\delta}\oplus\mathfrak{G}_{\delta+\gamma}\oplus\dots\oplus\mathfrak{G}_{\delta+n\gamma}$
is a representation of the $\mathfrak{sl}_{2}$ spanned by $e,h,f$, and since
$\mathfrak{G}_{\delta}$ and $\mathfrak{G}_{\delta+\gamma}$ are both nonzero,
the action of $e$ between them can not be 0. This implies (1).

(2) Assume that $[\mathfrak{N},\theta\mathfrak{N}]$ is not contained in
$\mathfrak{L}$. Then there are root vectors
$x\in\mathfrak{G}_{\alpha},\,y\in\mathfrak{G}_{\beta}$ in $\mathfrak{N}$ such
that $[x,\theta y]\notin\mathfrak{L}$. Since $[x,\theta
y]\in\mathfrak{G}_{\alpha-\beta}$, it follows that $\alpha-\beta$ is a root
either of $\mathfrak{N}$ or of $\mathfrak{N}^{-}$. If $\alpha-\beta$ is a root
of $\mathfrak{N}$, then since $\alpha=(\alpha-\beta)+\beta$ is a root, (1)
implies that $[\mathfrak{G}_{\alpha-\beta},\mathfrak{G}_{\beta}]\neq 0$, so
$[\mathfrak{N},\mathfrak{N}]\neq 0$ and $\mathfrak{N}$ is not abelian. If
$\alpha-\beta$ is a root of $\mathfrak{N}^{-}$, then $\beta-\alpha$ is a root
of $\mathfrak{N}$, so $\beta=(\beta-\alpha)+\alpha$ again implies that
$[\mathfrak{N},\mathfrak{N}]\neq 0$ and so $\mathfrak{N}$ is not abelian. ∎

The following theorem was proved in [TK68, §4]. See also [Kob08, Lemma 7.3.1]
and [RRS92]. We present a short proof for the convenience of the reader.

######  Theorem 2.2.

[TK68]. Let $\mathbb{G}$, $\mathbb{K}$ and $\mathbb{P}=\mathbb{L}\mathbb{N}$
be as above (i.e., as in Table 1). Then the following statements are
equivalent:

  1. (a)

$\mathbb{N}$ (or equivalently $\mathfrak{N}$) is abelian;

  2. (b)

$\mathbb{L}$ is a symmetric subgroup of $\mathbb{G}$;

  3. (c)

$\mathbb{P}\cap\mathbb{K}=\mathbb{L}\cap\mathbb{K}$ is a symmetric subgroup of
$\mathbb{K}$.

###### Proof.

$\mathbf{(a)\Rightarrow(b)}$. It is enough to show that in the decomposition
$\mathfrak{G}=\mathfrak{L}\oplus(\mathfrak{N}\oplus\theta\mathfrak{N})$ we
have
$[\mathfrak{N}\oplus\theta\mathfrak{N},\mathfrak{N}\oplus\theta\mathfrak{N}]\subseteq\mathfrak{L}$.
But Lemma 2.1(2) implies that

$[\mathfrak{N}\oplus\theta\mathfrak{N},\mathfrak{N}\oplus\theta\mathfrak{N}]=[\mathfrak{N},\mathfrak{N}]+[\mathfrak{N},\theta\mathfrak{N}]+[\theta\mathfrak{N},\mathfrak{N}]+[\theta\mathfrak{N},\theta\mathfrak{N}]=0+[\mathfrak{N},\theta\mathfrak{N}]+0\subseteq\mathfrak{L}.$

Note that the associated involution $\sigma$ is defined to be $+1$ on
$\mathfrak{L}$ and $(-1)$ on $\mathfrak{N}\oplus\theta\mathfrak{N}$. In
particular $\mathbb{P}$ is $\sigma$-stable.

$\mathbf{(b)\Rightarrow(c)}$. Let $\sigma$ be an involution of $\mathbb{G}$
such that
$\mathbb{G}^{\sigma}_{e}\subseteq\mathbb{L}\subseteq\mathbb{G}^{\sigma}$,
where $\mathbb{G}^{\sigma}_{e}$ denotes the connected component of
$\mathbb{G}^{\sigma}$. Since $\mathbb{P}$ is standard, we may assume $\sigma$
commutes with $\theta$. Then the restriction of $\sigma$ to $\mathbb{K}$ is an
involution, and
$\mathbb{K}^{\sigma}_{e}\subseteq\mathbb{L}\cap\mathbb{K}\subseteq\mathbb{K}^{\sigma}$.
So $\mathbb{L}\cap\mathbb{K}$ is a symmetric subgroup of $\mathbb{K}$. (We
remark that in all the examples we consider we will have
$\mathbb{L}=\mathbb{G}^{\sigma}$ and
$\mathbb{L}\cap\mathbb{K}=\mathbb{K}^{\sigma}$.)

$\mathbf{(c)\Rightarrow(a)}$. Suppose that $\mathfrak{N}$ is not abelian. Then
there are roots $\alpha,\beta$ of $\mathfrak{N}$ and
$x\in\mathfrak{G}_{\alpha}$, $y\in\mathfrak{G}_{\beta}$ such that $[x,y]\neq
0$. Then $x+\theta x,y+\theta
y\in(\mathfrak{N}\oplus\theta\mathfrak{N})^{\theta}$, and we have

$[x+\theta x,y+\theta y]=[x,y]+[x,\theta y]+[\theta x,y]+[\theta x,\theta y],$

with

$[x,y]\in\mathfrak{G}_{\alpha+\beta},\ [x,\theta
y]\in\mathfrak{G}_{\alpha-\beta},\ [\theta
x,y]\in\mathfrak{G}_{-\alpha+\beta},\ [\theta x,\theta
y]\in\mathfrak{G}_{-\alpha-\beta}.$

Since the root $\alpha+\beta$ is strictly greater than $\alpha-\beta$,
$-\alpha+\beta$ and $-\alpha-\beta$ (in the usual lexicographical order), we
see that $[x,y]\in\mathfrak{N}\setminus 0$ implies $[x+\theta x,y+\theta
y]\notin\mathfrak{L}\cap\mathfrak{K}$. It follows that
$\mathbb{L}\cap\mathbb{K}$ is not a symmetric subgroup of $\mathbb{K}$. ∎

######  Remark 2.3.

Suppose that $[\mathfrak{G},\mathfrak{G}]$ is simple and that
$\mathfrak{P}=\mathfrak{L}\oplus\mathfrak{N}$ is a standard parabolic
subalgebra as above. If $\mathfrak{N}$ is abelian, then $\mathfrak{P}$ is a
maximal parabolic subalgebra. See [RRS92, Lemma 2.2, p.651].

######  Proposition 2.4.

Let $\mathbb{G}$, $\mathbb{K}$ and $\mathbb{P}=\mathbb{L}\mathbb{N}$ be as
above (i.e., as in Table 1). Then $\mathbb{K}$ acts transitively on
$\mathbb{G}/\mathbb{P}$, and therefore $\mathbb{G}/\mathbb{P}$ is
diffeomorphic to
$\mathbb{K}/\mathbb{P}\cap\mathbb{K}=\mathbb{K}/\mathbb{L}\cap\mathbb{K}$. In
particular, $\mathbb{G}/\mathbb{P}$ is diffeomorphic to a symmetric space.

###### Proof.

It is clear from the Iwasawa decomposition that $\mathbb{K}$ acts transitively
on $\mathbb{G}/\mathbb{P}_{0}$, where $\mathbb{P}_{0}$ is a minimal parabolic
subgroup of $\mathbb{G}$ contained in $\mathbb{P}$. Since
$\mathbb{P}\supseteq\mathbb{P}_{0}$, there is a natural projection from
$\mathbb{G}/\mathbb{P}_{0}$ to $\mathbb{G}/\mathbb{P}$, sending
$g\mathbb{P}_{0}$ to $g\mathbb{P}$. This projection intertwines the
$\mathbb{G}$-actions, hence also the $\mathbb{K}$-actions. It follows that
$\mathbb{K}$ acts transitively on $\mathbb{G}/\mathbb{P}$. Indeed, if
$g\mathbb{P}\in\mathbb{G}/\mathbb{P}$, let $k\in\mathbb{K}$ be such that
$k\mathbb{P}_{0}=g\mathbb{P}_{0}$. Taking the projection we see that
$k\mathbb{P}=g\mathbb{P}$, which implies transitivity of the
$\mathbb{K}$-action. ∎

###  2.2. Ordinary Grassmannians

Let $\mathbb{F}$ be $\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$. Let
$\operatorname{Gr}_{p}(\mathbb{F}^{p+q})$ be the Grassmannian of
$p$-dimensional subspaces of the vector space $\mathbb{F}^{p+q}$. The group
$\mathbb{G}=\operatorname{GL}(p+q,\mathbb{F})$ clearly acts transitively on
$\operatorname{Gr}_{p}(\mathbb{F}^{p+q})$, so
$\operatorname{Gr}_{p}(\mathbb{F}^{p+q})=\mathbb{G}/\mathbb{P}$, where
$\mathbb{P}$ is the stabilizer in $\mathbb{G}$ of the standard $p$-dimensional
subspace

(2.5)
$\mathbb{F}^{p}=\\{(x_{1},\dots,x_{p},0,\dots,0)\,\big{|}\,x_{1},\dots,x_{p}\in\mathbb{F}\\}\subseteq\mathbb{F}^{p+q}.$

In other words,

$\mathbb{P}=\left\\{\begin{pmatrix}A&B\cr
0&C\end{pmatrix}\,\big{|}\,A\in\operatorname{GL}(p,\mathbb{F}),\,C\in\operatorname{GL}(q,\mathbb{F}),\,B\in
M_{pq}(\mathbb{F})\right\\}.$

Let $\sigma$ be the involution of $\mathbb{G}$ defined as in (1.3), i.e.,
$\sigma(g)=I_{p,q}gI_{p,q}$ where $I_{p,q}=\left(\begin{smallmatrix}I_{p}&0\cr
0&-I_{q}\end{smallmatrix}\right)$. Then $\mathbb{G}^{\sigma}$ is equal to the
Levi subgroup $\mathbb{L}$ of $\mathbb{P}$ and it consists of block diagonal
matrices in $\mathbb{G}$, i.e.,

$\mathbb{G}^{\sigma}=\mathbb{L}=\operatorname{GL}(p,\mathbb{F})\times\operatorname{GL}(q,\mathbb{F}).$

The maximal compact subgroup $\mathbb{K}$ of $\mathbb{G}$ is the unitary group
of $\mathbb{F}^{p+q}$ with respect to the standard inner product, denoted as
$\operatorname{U}(p+q,\mathbb{F})$. In other words, $\mathbb{K}$ is
$\operatorname{O}(p+q)$ if $\mathbb{F}=\mathbb{R}$, $\operatorname{U}(p+q)$ if
$\mathbb{F}=\mathbb{C}$, and $\operatorname{Sp}(p+q)$ if
$\mathbb{F}=\mathbb{H}$. $\mathbb{P}\cap\mathbb{K}=\mathbb{K}^{\sigma}$ is
$\operatorname{U}(p,\mathbb{F})\times\operatorname{U}(q,\mathbb{F})$, embedded
block diagonally. In other words, $\mathbb{K}^{\sigma}$ is
$\operatorname{O}(p)\times\operatorname{O}(q)$ if $\mathbb{F}=\mathbb{R}$,
$\operatorname{U}(p)\times\operatorname{U}(q)$ if $\mathbb{F}=\mathbb{C}$, and
$\operatorname{Sp}(p)\times\operatorname{Sp}(q)$ if $\mathbb{F}=\mathbb{H}$.

Now Proposition 2.4 implies the following well known result which can be found
for example in [Oni94, Ch. I, §4].

######  Proposition 2.6.

$\operatorname{Gr}_{p}(\mathbb{F}^{p+q})=\mathbb{G}/\mathbb{P}$ is
diffeomorphic to
$\operatorname{U}(p+q,\mathbb{F})/\operatorname{U}(p,\mathbb{F})\times\operatorname{U}(q,\mathbb{F})$.
In other words, $\operatorname{Gr}_{p}(\mathbb{R}^{p+q})$ is diffeomorphic to
$\operatorname{O}(p+q)/\operatorname{O}(p)\times\operatorname{O}(q)$,
$\operatorname{Gr}_{p}(\mathbb{C}^{p+q})$ is diffeomorphic to
$\operatorname{U}(p+q)/\operatorname{U}(p)\times\operatorname{U}(q)$, and
$\operatorname{Gr}_{p}(\mathbb{H}^{p+q})$ is diffeomorphic to
$\operatorname{Sp}(p+q)/\operatorname{Sp}(p)\times\operatorname{Sp}(q)$

For cohomology computation we need $G=\mathbb{K}$ to be connected, and it is
connected if $\mathbb{F}$ is $\mathbb{C}$ or $\mathbb{H}$. If
$\mathbb{F}=\mathbb{R}$, we note that

$\mathbb{G}/\mathbb{P}\cong\operatorname{SO}(p+q)/S(\operatorname{O}(p)\times\operatorname{O}(q)).$

This follows immediately from the fact that
$\operatorname{SL}(p+q,\mathbb{R})$ acts transitively on
$\operatorname{Gr}_{p}(\mathbb{R}^{p+q})$. We can now conclude

######  Corollary 2.7.

The cohomology ring (with complex coefficients) of the Grassmannian
$\operatorname{Gr}_{p}(\mathbb{F}^{p+q})$ is described by: Theorem 4.21 if
$\mathbb{F}=\mathbb{R}$; Theorem 4.5 if $\mathbb{F}=\mathbb{C}$; Theorem 4.19
if $\mathbb{F}=\mathbb{H}$.

Since the involution $\sigma\theta$ of $\mathbb{G}$ is given by (1.4), i.e.,
by $\sigma\theta(g)=I_{p,q}(\bar{g}^{t})^{-1}I_{p,q}$, the group
$G_{\mathbb{R}}=\mathbb{G}^{\sigma\theta}$ is equal to
$\operatorname{U}(p,q;\mathbb{F})$. In other words, $G_{\mathbb{R}}$ is
$\operatorname{O}(p,q)$ if $\mathbb{F}=\mathbb{R}$; $\operatorname{U}(p,q)$ if
$\mathbb{F}=\mathbb{C}$; and $\operatorname{Sp}(p,q)$ if
$\mathbb{F}=\mathbb{H}$.

###  2.3. The symplectic Lagrangian Grassmannians

Let $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$, and let
$\operatorname{LGr}(\mathbb{R}^{2n})$ be the (symplectic) Lagrangian
Grassmannian, i.e., the manifold of all Lagrangian subspaces of
$\mathbb{F}^{2n}$ with respect to the standard symplectic form
$\langle\,,\rangle$ given by

$\langle x,y\rangle=x^{t}J_{n}y,\qquad\text{where}\quad
J_{n}=\begin{pmatrix}0&I_{n}\cr-I_{n}&0\end{pmatrix}.$

Let $\mathbb{G}$ be the group $\operatorname{Sp}(2n,\mathbb{F})$ of $2n\times
2n$ matrices over $\mathbb{F}$ preserving the form $\langle\,,\rangle$, i.e.,
satisfying $g^{t}J_{n}g=J_{n}$. Then $\mathbb{G}$ acts on
$\operatorname{LGr}(\mathbb{F}^{2n})$, and this action is transitive by Witt’s
Theorem 1.1. Thus $\operatorname{LGr}(\mathbb{F}^{2n})=\mathbb{G}/\mathbb{P}$,
where $\mathbb{P}$ is the (Siegel) parabolic subgroup of
$\operatorname{Sp}(2n,\mathbb{F})$, defined as the stabilizer of the standard
Lagrangian subspace

$L_{0}=\\{(x_{1},\dots,x_{n},0,\dots,0)\,\big{|}\,x_{1},\dots,x_{n}\in\mathbb{F}\\}\subseteq\mathbb{F}^{2n}.$

Writing $g\in\mathbb{G}$ as a block matrix $\left(\begin{smallmatrix}A&B\cr
C&D\end{smallmatrix}\right)$ with $n\times n$ blocks, the condition
$g^{t}J_{n}g=J_{n}$ implies

(2.8)
$\mathbb{G}=\operatorname{Sp}(2n,\mathbb{F})=\left\\{\begin{pmatrix}A&B\cr
C&D\end{pmatrix}\in\operatorname{GL}(2n,\mathbb{F})\,\big{|}\,A^{t}C=C^{t}A,\
B^{t}D=D^{t}B,\ A^{t}D-C^{t}B=I_{n}\right\\}.$

It follows that

(2.9) $\mathbb{P}=\left\\{\begin{pmatrix}A&B\cr
0&D\end{pmatrix}\in\operatorname{GL}(2n,\mathbb{F})\,\big{|}\,B^{t}D=D^{t}B,\
A^{t}D=I_{n}\right\\}.$

Let $\sigma$ be the involution of $\mathbb{G}$ given by (1.3), i.e., by
$\sigma(g)=I_{n,n}gI_{n,n}$. Then $\mathbb{G}^{\sigma}$ is equal to the Levi
subgroup $\mathbb{L}$ of $\mathbb{P}$ and it consists of block diagonal
matrices in $\mathbb{G}$, i.e.,

$\mathbb{G}^{\sigma}=\mathbb{L}=\left\\{\begin{pmatrix}A&0\cr
0&(A^{t})^{-1}\end{pmatrix}\,\big{|}\,A\in\operatorname{GL}(n,\mathbb{F})\right\\}\cong\operatorname{GL}(n,\mathbb{F})\quad\text{via}\
\begin{pmatrix}A&0\cr 0&(A^{t})^{-1}\end{pmatrix}\leftrightarrow A.$

Since $\mathbb{K}$ consists of the fixed points of the Cartan involution
$\theta(g)=(\bar{g}^{t})^{-1}$, and since $g\in\mathbb{G}$ is equivalent to
$(g^{t})^{-1}=J_{n}gJ_{n}^{-1}$, we see that $\theta(g)=g$ is equivalent to
$J_{n}\bar{g}=gJ_{n}$. This implies

(2.10) $\mathbb{K}=\left\\{\begin{pmatrix}A&-\bar{C}\cr
C&\bar{A}\end{pmatrix}\,\big{|}\,A^{t}C=C^{t}A,\
\bar{A}^{t}A+\bar{C}^{t}C=I_{n}\right\\}.$

If $\mathbb{F}=\mathbb{R}$ (so the bars can be omitted), this is exactly the
standard description of $\operatorname{U}(n)$ inside
$\operatorname{GL}(2n,\mathbb{R})$; more precisely,
$\mathbb{K}\cong\operatorname{U}(n)$ via $\left(\begin{smallmatrix}A&-C\cr
C&A\end{smallmatrix}\right)\leftrightarrow A+iC$.

If $\mathbb{F}=\mathbb{C}$, then (2.10) is exactly the standard description of
$\operatorname{Sp}(n)$ inside $\operatorname{GL}(2n,\mathbb{C})$.

We now also see that

$\mathbb{P}\cap\mathbb{K}=\mathbb{K}^{\sigma}=\left\\{\begin{pmatrix}A&0\cr
0&\bar{A}\end{pmatrix}\,\big{|}\,A^{t}A=I\right\\}\cong\operatorname{U}(n,\mathbb{F})\quad\text{via}\
\begin{pmatrix}A&0\cr 0&\bar{A}\end{pmatrix}\leftrightarrow A.$

In other words, if $\mathbb{F}=\mathbb{R}$, then
$\mathbb{K}^{\sigma}=\operatorname{O}(n)$ via $\left(\begin{smallmatrix}A&0\cr
0&A\end{smallmatrix}\right)\leftrightarrow A$, and if $\mathbb{F}=\mathbb{C}$,
then $\mathbb{K}^{\sigma}=\operatorname{U}(n)$ via
$\left(\begin{smallmatrix}A&0\cr
0&\bar{A}\end{smallmatrix}\right)\leftrightarrow A$.

Now Proposition 2.4 implies

######  Proposition 2.11.

The real symplectic Lagrangian Grassmannian
$\operatorname{LGr}(\mathbb{R}^{2n})$ is diffeomorphic to
$\operatorname{U}(n)/\operatorname{O}(n)$, while the complex symplectic
Lagrangian Grassmannian $\operatorname{LGr}(\mathbb{C}^{2n})$ is diffeomorphic
to $\operatorname{Sp}(n)/\operatorname{U}(n)$.

######  Corollary 2.12.

The cohomology ring (with complex coefficients) of the symplectic Lagrangian
Grassmannian $\operatorname{LGr}(\mathbb{F}^{2n})$ is described by: Theorem
3.29 if $\mathbb{F}=\mathbb{R}$; Theorem 4.25 if $\mathbb{F}=\mathbb{C}$.

Finally, we describe the group $\mathbb{G}^{\sigma\theta}$. Let first
$\mathbb{F}=\mathbb{R}$. Then since
$\sigma\theta(g)=I_{n,n}(g^{t})^{-1}I_{n,n}$, and since any $g\in\mathbb{G}$
satisfies $(g^{t})^{-1}=J_{n}gJ_{n}^{-1}$, $\sigma\theta(g)=g$ is equivalent
to $gD_{n}=D_{n}g$ where $D_{n}=\left(\begin{smallmatrix}0&I_{n}\cr
I_{n}&0\end{smallmatrix}\right)$. It follows that

$\mathbb{G}^{\sigma\theta}=\left\\{g=\begin{pmatrix}A&B\cr
B&A\end{pmatrix}\,\big{|}\,A^{t}B=B^{t}A,\,A^{t}A-B^{t}B=I_{n}\right\\}.$

Conjugating $\left(\begin{smallmatrix}A&B\cr B&A\end{smallmatrix}\right)$ by
$\left(\begin{smallmatrix}I_{n}&I_{n}\cr I_{n}&-I_{n}\end{smallmatrix}\right)$
we get the matrix $\left(\begin{smallmatrix}A+B&0\cr
0&A-B\end{smallmatrix}\right)$, and the conditions
$A^{t}B=B^{t}A,\,A^{t}A-B^{t}B=I_{n}$ imply $(A+B)^{t}(A-B)=I_{n}$, so
$A-B=((A+B)^{t})^{-1}$. Conversely, starting from the matrix
$\left(\begin{smallmatrix}Z&0\cr 0&(Z^{t})^{-1}\end{smallmatrix}\right)$ and
setting $A=\frac{1}{2}(Z+(Z^{t})^{-1})$, $B=\frac{1}{2}(Z-(Z^{t})^{-1})$, we
get $A^{t}B=B^{t}A,\,A^{t}A-B^{t}B=I_{n}$. Thus

$\mathbb{G}^{\sigma\theta}\cong\left\\{\begin{pmatrix}Z&0\cr
0&(Z^{t})^{-1}\end{pmatrix}\,\big{|}\,Z\in\operatorname{GL}(n,\mathbb{R})\right\\}\cong\operatorname{GL}(n,\mathbb{R})\
\text{via}\ \begin{pmatrix}Z&0\cr 0&(Z^{t})^{-1}\end{pmatrix}\leftrightarrow
Z.$

Now let $\mathbb{F}=\mathbb{C}$. Since
$\sigma\theta(g)=I_{n,n}(\bar{g}^{t})^{-1}I_{n,n}$, and since any
$g\in\mathbb{G}$ satisfies $(g^{t})^{-1}=J_{n}gJ_{n}^{-1}$, we see that
$\sigma\theta(g)=D_{n}\bar{g}D_{n}$. We claim that
$\mathbb{G}^{\sigma\theta}\cong\operatorname{Sp}(2n,\mathbb{R})$. To see this,
we note that

(2.13)
$C_{n}=\frac{1}{2}\begin{pmatrix}(1+i)I_{n}&(1-i)I_{n}\cr(1-i)I_{n}&(1+i)I_{n}\end{pmatrix}\qquad\text{implies}\quad
C_{n}^{2}=D_{n}\text{ and }C_{n}^{-1}=\bar{C}_{n}.$

This implies that

$g\in\mathbb{G}^{\sigma\theta}\qquad\text{if and only
if}\qquad\overline{C_{n}gC_{n}^{-1}}=C_{n}gC_{n}^{-1},$

i.e., that
$\mathbb{G}^{\sigma\theta}=C_{n}^{-1}\operatorname{Sp}(2n,\mathbb{R})C_{n}\cong\operatorname{Sp}(2n,\mathbb{R})$.

###  2.4. Orthogonal Lagrangian Grassmannians

Let $\operatorname{OLGr}(\mathbb{C}^{2n})$ be the (complex) orthogonal
Lagrangian Grassmannian. In other words,
$\operatorname{OLGr}(\mathbb{C}^{2n})$ is the manifold of all Lagrangian
subspaces of $\mathbb{C}^{2n}$ with respect to the symmetric bilinear form
$\langle\,,\rangle$ defined by

$\langle
x,y\rangle=\sum_{r=1}^{n}x_{r}y_{n+r}+\sum_{r=1}^{n}x_{n+r}y_{r}=x^{t}D_{n}y,$

where as before, $D_{n}=\left(\begin{smallmatrix}0&I_{n}\cr
I_{n}&0\end{smallmatrix}\right)$.

Let $\mathbb{G}=\operatorname{O}(2n,\mathbb{C})$ be the group of $2n\times 2n$
complex matrices preserving the form $\langle\,,\rangle$, i.e., satisfying
$g^{t}D_{n}g=D_{n}$. Writing $g=\left(\begin{smallmatrix}A&B\cr
C&D\end{smallmatrix}\right)$ we see that

(2.14)
$\mathbb{G}=\operatorname{O}(2n,\mathbb{C})=\left\\{\begin{pmatrix}A&B\cr
C&D\end{pmatrix}\in\operatorname{GL}(2n,\mathbb{C})\,\big{|}\,C^{t}A=-A^{t}C,\,D^{t}B=-B^{t}D,\,A^{t}D+C^{t}B=I_{n}\right\\}.$

The group $\mathbb{G}$ acts on $\operatorname{OLGr}(\mathbb{C}^{2n})$, and
this action is transitive by Witt’s Theorem 1.1. Thus
$\operatorname{OLGr}(\mathbb{C}^{2n})=\mathbb{G}/\mathbb{P}$, where
$\mathbb{P}$ is the parabolic subgroup of $\mathbb{G}$, defined as the
stabilizer of the standard Lagrangian subspace

$L_{0}=\\{(x_{1},\dots,x_{n},0,\dots,0)\,\big{|}\,x_{1},\dots,x_{n}\in\mathbb{C}\\}\subset\mathbb{C}^{2n}.$

An element $g=\left(\begin{smallmatrix}A&B\cr C&D\end{smallmatrix}\right)$ of
$\mathbb{G}$ stabilizes $L_{0}$ if and only if $C=0$, so

(2.15) $\mathbb{P}=\left\\{\begin{pmatrix}A&B\cr
0&D\end{pmatrix}\in\operatorname{GL}(2n,\mathbb{C})\,\big{|}\,D^{t}B=-B^{t}D,\,A^{t}D=I_{n}\right\\}.$

Let $\sigma$ be the involution of $\mathbb{G}$ defined by (1.3), i.e., by
$\sigma(g)=I_{n,n}gI_{n,n}$. Then $\mathbb{G}^{\sigma}$ is equal to the Levi
subgroup $\mathbb{L}$ of $\mathbb{P}$ and it consists of block diagonal
matrices in $\mathbb{G}$, i.e.,

$\mathbb{G}^{\sigma}=\mathbb{L}=\left\\{\begin{pmatrix}A&0\cr
0&(A^{t})^{-1}\end{pmatrix}\,\big{|}\,A\in\operatorname{GL}(n,\mathbb{C})\right\\}\cong\operatorname{GL}(n,\mathbb{C})\quad\text{via}\
\begin{pmatrix}A&0\cr 0&(A^{t})^{-1}\end{pmatrix}\leftrightarrow A.$

Since $\theta(g)=(\bar{g}^{t})^{-1}$ and since $g\in\mathbb{G}$ is equivalent
to $(g^{t})^{-1}=D_{n}gD_{n}$, we see that $\theta(g)=g$ is equivalent to
$D_{n}\bar{g}D_{n}=g$, or $D_{n}\bar{g}=gD_{n}$. Thus

(2.16) $\mathbb{K}=\left\\{\begin{pmatrix}A&\bar{C}\cr
C&\bar{A}\end{pmatrix}\in\operatorname{GL}(2n,\mathbb{C})\,\big{|}\,C^{t}A=-A^{t}C,\
\bar{A}^{t}A+\bar{C}^{t}C=I_{n}\right\\}.$

To identify this subgroup, we connect our $\mathbb{G}$ with the more usual
group $\mathbb{G}^{\prime}=\operatorname{O}(2n,\mathbb{C})^{\prime}$ given by
$g^{t}g=I_{2n}$. The maximal compact subgroup $\mathbb{K}^{\prime}$ of
$\mathbb{G}^{\prime}$ is given by the condition $(\bar{g}^{t})^{-1}=g$, or
equivalently $\bar{g}=g$, so $\mathbb{K}^{\prime}=\operatorname{O}(2n)$. Since
$\mathbb{G}$ and $\mathbb{G}^{\prime}$ are isomorphic,
$\mathbb{K}\cong\operatorname{O}(2n)$. In fact, explicit isomorphisms
$\mathbb{G}\cong\mathbb{G}^{\prime}$ and $\mathbb{K}\cong\mathbb{K}^{\prime}$
are given by conjugation by the matrix $C_{n}$ of (2.13).

We also see that

$\mathbb{P}\cap\mathbb{K}=\mathbb{K}^{\sigma}=\left\\{\begin{pmatrix}A&0\cr
0&\bar{A}\end{pmatrix}\,\big{|}\,\bar{A}^{t}A=I_{n}\right\\}\cong\operatorname{U}(n)\quad\text{via}\quad\begin{pmatrix}A&0\cr
0&\bar{A}\end{pmatrix}\leftrightarrow A.$

Now Proposition 2.4 implies

######  Proposition 2.17.

The orthogonal Lagrangian Grassmannian $\operatorname{OLGr}(\mathbb{C}^{2n})$
is diffeomorphic to $\operatorname{O}(2n)/\operatorname{U}(n)$.

Since our computation of cohomology of a compact symmetric space $G/K$
requires $G$ to be connected, we replace
$\operatorname{O}(2n)/\operatorname{U}(n)$ by
$\operatorname{SO}(2n)/\operatorname{U}(n)$. The orbit of
$\operatorname{SO}(2n)$ on $\operatorname{OLGr}(\mathbb{C}^{2n})$ is one of
the two components of $\operatorname{OLGr}(\mathbb{C}^{2n})$, which we denote
by $\operatorname{OLGr}^{+}(\mathbb{C}^{2n})$, and still call it the
orthogonal Lagrangian Grassmannian. The other component of
$\operatorname{OLGr}(\mathbb{C}^{2n})$ is diffeomorphic to
$\operatorname{OLGr}^{+}(\mathbb{C}^{2n})$ and thus has the same cohomology.

######  Corollary 2.18.

The cohomology ring (with complex coefficients) of the orthogonal Lagrangian
Grassmannian
$\operatorname{OLGr}^{+}(\mathbb{C}^{2n})\cong\operatorname{SO}(2n)/\operatorname{U}(n)$
is described by Theorem 4.28.

Finally, we describe the group $G_{\mathbb{R}}=\mathbb{G}^{\sigma\theta}$ in
case $\mathbb{G}=\operatorname{SO}(2n,\mathbb{C})$. Since
$\sigma\theta(g)=I_{n,n}(\bar{g}^{t})^{-1}I_{n,n}$, we see that
$\mathbb{G}^{\sigma\theta}=\operatorname{O}(2n,\mathbb{C})\cap\operatorname{U}(n,n)=\operatorname{SO}(2n,\mathbb{C})\cap\operatorname{SU}(n,n)$,
and this is exactly the description of $\operatorname{SO}^{*}(2n)$ given in
[Kna96, 1.141].

######  Remark 2.19.

One could define the group $\operatorname{O}^{*}(2n)$ as
$\operatorname{O}(2n,\mathbb{C})\cap U(n,n)$, or alternatively, as the group
of automorphisms of $\mathbb{H}^{n}$ preserving a skew Hermitian form; see
Section 2.6. Conceivably, an element of this group could have determinant
equal to $\pm 1$. However, we prove in Section 2.6 that the maximal compact
subgroup of this group is $\operatorname{U}(n)$, so it follows that the group
is connected and the determinant must be 1. In other words,
$\operatorname{O}^{*}(2n)=\operatorname{SO}^{*}(2n)$.

###  2.5. The Hermitian Lagrangian Grassmannians

Let $\mathbb{F}$ be $\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$ and let
$\operatorname{HLGr}(\mathbb{F}^{2n})$ be the Hermitian Lagrangian
Grassmannian. In other words, $\operatorname{HLGr}(\mathbb{F}^{2n})$ is the
manifold of all Lagrangian subspaces of $\mathbb{F}^{2n}$ with respect to the
Hermitian form $\langle\,,\rangle$ of signature $(n,n)$, defined by

$\langle
x,y\rangle=\sum_{r=1}^{n}\bar{x}_{r}y_{n+r}+\sum_{r=1}^{n}\bar{x}_{n+r}y_{r}=\bar{x}^{t}D_{n}y,$

where as before, $D_{n}=\left(\begin{smallmatrix}0&I_{n}\cr
I_{n}&0\end{smallmatrix}\right)$.

Let $\mathbb{G}=\operatorname{U}(n,n;\mathbb{F})$ be the group of $2n\times
2n$ matrices over $\mathbb{F}$ preserving the form $\langle\,,\rangle$, i.e.,
satisfying $\bar{g}^{t}D_{n}g=D_{n}$. So if $\mathbb{F}=\mathbb{R}$,
$\mathbb{G}=\operatorname{O}(n,n)$; if $\mathbb{F}=\mathbb{C}$,
$\mathbb{G}=\operatorname{U}(n,n)$; and if $\mathbb{F}=\mathbb{H}$,
$\mathbb{G}=\operatorname{Sp}(n,n)$.

Writing $g=\left(\begin{smallmatrix}A&B\cr C&D\end{smallmatrix}\right)$ with
$n\times n$ blocks, we see that

(2.20)
$\mathbb{G}=\operatorname{U}(n,n;\mathbb{F})=\left\\{\begin{pmatrix}A&B\cr
C&D\end{pmatrix}\in\operatorname{GL}(2n,\mathbb{F})\,\big{|}\,\bar{C}^{t}A=-\bar{A}^{t}C,\,\bar{D}^{t}B=-\bar{B}^{t}D,\,\bar{A}^{t}D+\bar{C}^{t}B=I_{n}\right\\}.$

The group $\mathbb{G}$ acts on $\operatorname{HLGr}(\mathbb{F}^{2n})$, and
this action is transitive by Witt’s Theorem 1.1. Thus
$\operatorname{HLGr}(\mathbb{F}^{2n})=\mathbb{G}/\mathbb{P}$, where
$\mathbb{P}$ is the parabolic subgroup of $\mathbb{G}$ defined as the
stabilizer of the standard Lagrangian subspace

$L_{0}=\\{(x_{1},\dots,x_{n},0,\dots,0)\,\big{|}\,x_{1},\dots,x_{n}\in\mathbb{F}\\}\subset\mathbb{F}^{2n}.$

It follows that

(2.21) $\mathbb{P}=\left\\{\begin{pmatrix}A&B\cr
0&D\end{pmatrix}\in\operatorname{GL}(2n,\mathbb{F})\,\big{|}\,\bar{D}^{t}B=-\bar{B}^{t}D,\,\bar{A}^{t}D=I_{n}\right\\}.$

Let $\sigma$ be an involution of $\mathbb{G}$ defined by (1.3), i.e., by
$\sigma(g)=I_{n,n}gI_{n,n}$. Then $\mathbb{G}^{\sigma}$ is equal to the Levi
subgroup $\mathbb{L}$ of $\mathbb{P}$ and it consists of block diagonal
matrices in $\mathbb{G}$, i.e.,

$\mathbb{G}^{\sigma}=\mathbb{L}=\left\\{\begin{pmatrix}A&0\cr
0&(\bar{A}^{t})^{-1}\end{pmatrix}\,\big{|}\,A\in\operatorname{GL}(n,\mathbb{F})\right\\}\cong\operatorname{GL}(n,\mathbb{F})\quad\text{via}\
\begin{pmatrix}A&0\cr 0&(\bar{A}^{t})^{-1}\end{pmatrix}\leftrightarrow A.$

Since the Cartan involution is $\theta(g)=(\bar{g}^{t})^{-1}$ and since
$g\in\mathbb{G}$ is equivalent to $(\bar{g}^{t})^{-1}=D_{n}gD_{n}$, we see
that $\theta(g)=g$ is equivalent to $D_{n}gD_{n}=g$, or $D_{n}g=gD_{n}$. It
follows that

(2.22) $\mathbb{K}=\left\\{\begin{pmatrix}A&B\cr
B&A\end{pmatrix}\in\operatorname{GL}(2n,\mathbb{F})\,\big{|}\,\bar{B}^{t}A=-\bar{A}^{t}B,\
\bar{A}^{t}A+\bar{B}^{t}B=I_{n}\right\\}.$

To identify this subgroup, we conjugate the matrix
$\left(\begin{smallmatrix}A&B\cr B&A\end{smallmatrix}\right)$ by
$\left(\begin{smallmatrix}I_{n}&I_{n}\cr I_{n}&-I_{n}\end{smallmatrix}\right)$
and get the matrix $\left(\begin{smallmatrix}A+B&0\cr
0&A-B\end{smallmatrix}\right)$. The conditions
$\bar{B}^{t}A=-\bar{A}^{t}B,\,\bar{A}^{t}A+\bar{B}^{t}B=I_{n}$ imply
$\overline{(A+B)}^{t}(A+B)=I_{n}$ and $\overline{(A-B)}^{t}(A-B)=I_{n}$, so
$A+B$ and $A-B$ are in $\operatorname{U}(n,\mathbb{F})$ (i.e., in
$\operatorname{O}(n)$ if $\mathbb{F}=\mathbb{R}$; in $\operatorname{U}(n)$ if
$\mathbb{F}=\mathbb{C}$; and in $\operatorname{Sp}(n)$ if
$\mathbb{F}=\mathbb{H}$). Conversely, starting from matrices $Z$ and $W$ in
$\operatorname{U}(n,\mathbb{F})$, we can reconstruct $A$ and $B$ as
$A=\frac{1}{2}(Z+W)$, $B=\frac{1}{2}(Z-W)$, and the matrix
$\left(\begin{smallmatrix}A&B\cr B&A\end{smallmatrix}\right)$ will satisfy the
conditions $\bar{B}^{t}A=-\bar{A}^{t}B,\,\bar{A}^{t}A+\bar{B}^{t}B=I_{n}$. So
we found an explicit isomorphism
$\mathbb{K}\cong\operatorname{U}(n,\mathbb{F})\times\operatorname{U}(n,\mathbb{F})$.
Under this isomorphism, the subgroup
$\mathbb{P}\cap\mathbb{K}=\mathbb{K}^{\sigma}$ corresponds to the diagonal
$\Delta\operatorname{U}(n,\mathbb{F})\subset\operatorname{U}(n,\mathbb{F})\times\operatorname{U}(n,\mathbb{F})$.

Now Proposition 2.4 implies

######  Proposition 2.23.

The Hermitian Lagrangian Grassmannian $\operatorname{HLGr}(\mathbb{F}^{2n})$
is diffeomorphic to
$\operatorname{U}(n,\mathbb{F})\times\operatorname{U}(n,\mathbb{F})/\Delta\operatorname{U}(n,\mathbb{F})$.
In other words, $\operatorname{HLGr}(\mathbb{F}^{2n})$ is diffeomorphic to
$\operatorname{O}(n)\times\operatorname{O}(n)/\Delta\operatorname{O}(n)$ if
$\mathbb{F}=\mathbb{R}$; to
$\operatorname{U}(n)\times\operatorname{U}(n)/\Delta\operatorname{U}(n)$ if
$\mathbb{F}=\mathbb{C}$; and to
$\operatorname{Sp}(n)\times\operatorname{Sp}(n)/\Delta\operatorname{Sp}(n)$ if
$\mathbb{F}=\mathbb{H}$ .

To compute the cohomology of $G/K$ we need $G$ to be connected, and in case
$\mathbb{F}=\mathbb{R}$ the group
$G=\mathbb{K}=\operatorname{O}(n)\times\operatorname{O}(n)$ is not connected.
Thus in the real case we replace
$\operatorname{O}(n)\times\operatorname{O}(n)/\Delta\operatorname{O}(n)$ by
$\operatorname{SO}(n)\times\operatorname{SO}(n)/\Delta\operatorname{SO}(n)$.
This amounts to replacing $\operatorname{HLGr}(\mathbb{R}^{2n})$ by the orbit
$\operatorname{HLGr}^{+}(\mathbb{R}^{2n})$ of
$\operatorname{SO}(n)\times\operatorname{SO}(n)$ which is one of the two
connected components of $\operatorname{HLGr}(\mathbb{R}^{2n})$.

######  Corollary 2.24.

The cohomology rings (with complex coefficients) of the Hermitian Lagrangian
Grassmannians $\operatorname{HLGr}^{+}(\mathbb{R}^{2n})$,
$\operatorname{HLGr}(\mathbb{C}^{2n})$, and
$\operatorname{HLGr}(\mathbb{H}^{2n})$, are described by Theorem 4.30.

Finally, we describe the group $G_{\mathbb{R}}=\mathbb{G}^{\sigma\theta}$.
Since $\sigma\theta(g)=I_{n,n}(\bar{g}^{t})^{-1}I_{n,n}$ and since any
$g\in\mathbb{G}$ satisfies $(\bar{g}^{t})^{-1}=D_{n}gD_{n}$, we see that

$\sigma\theta(g)=I_{n,n}D_{n}gD_{n}I_{n,n}=J_{n}gJ_{n}^{-1}.$

Thus $\sigma\theta(g)=g$ is equivalent to $J_{n}g=gJ_{n}$, and it follows that

(2.25) $\mathbb{G}^{\sigma\theta}=\left\\{\begin{pmatrix}A&-C\cr
C&A\end{pmatrix}\,\big{|}\,\bar{C}^{t}A=-\bar{A}^{t}C,\,\bar{A}^{t}A-\bar{C}^{t}C=I_{n}\right\\}.$

If $\mathbb{F}=\mathbb{R}$, recall that
$A+iC\mapsto\left(\begin{smallmatrix}A&-C\cr C&A\end{smallmatrix}\right)$ is
the standard embedding of $\operatorname{GL}(n,\mathbb{C})$ into
$\operatorname{GL}(2n,\mathbb{R})$, and note that the conditions
$C^{t}A=-A^{t}C,\,A^{t}A-C^{t}C=I_{n}$ correspond to $(A+iC)^{t}(A+iC)=I_{n}$.
It follows that $\mathbb{G}^{\sigma\theta}=\operatorname{O}(n,\mathbb{C})$.

If $\mathbb{F}=\mathbb{C}$, we conjugate $\left(\begin{smallmatrix}A&-C\cr
C&A\end{smallmatrix}\right)$ by $\left(\begin{smallmatrix}1&i\cr
1&-i\end{smallmatrix}\right)$ and get $\left(\begin{smallmatrix}A+iC&0\cr
0&A-iC\end{smallmatrix}\right)$. The conditions
$\bar{C}^{t}A=-\bar{A}^{t}C,\,\bar{A}^{t}A-\bar{C}^{t}C=I_{n}$ imply
$\overline{(A+iC})^{t}(A-iC)=I_{n}$. Conversely, given
$\left(\begin{smallmatrix}Z&0\cr 0&(\bar{Z}^{t})^{-1}\end{smallmatrix}\right)$
we can reconstruct $A$ and $C$ as $A=\frac{1}{2}(Z+(\bar{Z}^{t})^{-1})$ and
$C=\frac{1}{2i}(Z-(\bar{Z}^{t})^{-1})$ and get
$\left(\begin{smallmatrix}A&-C\cr
C&A\end{smallmatrix}\right)\in\mathbb{G}^{\sigma}$. It follows that

$\mathbb{G}^{\sigma\theta}\cong\operatorname{GL}(n,\mathbb{C}),\quad\text{via}\
\begin{pmatrix}A&-C\cr C&A\end{pmatrix}\leftrightarrow\begin{pmatrix}A+iC&0\cr
0&A-iC\end{pmatrix}\leftrightarrow A+iC.$

If $\mathbb{F}=\mathbb{H}$, we claim that $\mathbb{G}^{\sigma\theta}$ is
isomorphic to $\operatorname{Sp}(2n,\mathbb{C})$. To see this, we consider the
map

$\begin{pmatrix}A&-C\cr
C&A\end{pmatrix}=\begin{pmatrix}A_{1}+jA_{2}&-C_{1}-jC_{2}\cr
C_{1}+jC_{2}&A_{1}+jA_{2}\end{pmatrix}\mapsto\begin{pmatrix}A_{1}+iC_{1}&-\bar{A}_{2}-i\bar{C}_{2}\cr
A_{2}+iC_{2}&\bar{A}_{1}+i\bar{C}_{1}\end{pmatrix}.$

It is a tedious but straightforward computation to check that the conditions
(2.25) imply the conditions in (2.8), so our map sends
$\mathbb{G}^{\sigma\theta}$ into $\operatorname{Sp}(2n,\mathbb{C})$.
Conversely, if $\left(\begin{smallmatrix}X&Y\cr
Z&T\end{smallmatrix}\right)\in\operatorname{Sp}(2n,\mathbb{C})$, then we can
reconstruct an element of $\mathbb{G}^{\sigma\theta}$ mapping to
$\left(\begin{smallmatrix}X&Y\cr Z&T\end{smallmatrix}\right)$; it is given by

$A_{1}=\frac{1}{2}(X+\bar{T}),\quad A_{2}=\frac{1}{2}(Z-\bar{Y}),\quad
C_{1}=\frac{1}{2i}(X-\bar{T}),\quad C_{2}=\frac{1}{2i}(Z+\bar{Y}).$

(Another tedious computation shows that this element does satisfy the
conditions of (2.25).)

###  2.6. The skew Hermitian quaternionic Lagrangian Grassmannians

Consider the skew Hermitian form on $\mathbb{H}^{2n}$ given by

(2.26)
$(x\,|\,y)=\bar{x}^{t}J_{n}y=\sum_{r=1}^{n}\bar{x}_{r}y_{n+r}-\sum_{r=1}^{n}\bar{x}_{n+r}y_{r},$

where as before, $J_{n}=\left(\begin{smallmatrix}0&I_{n}\cr-
I_{n}&0\end{smallmatrix}\right)$. (Note that the form $(\,|\,)$ is different
from, but equivalent to, the form considered in [Ros02]. Also, recall that
$\mathbb{H}$ acts on $\mathbb{H}^{2n}$ by right scalar multiplication, and
that the form $(\,|\,)$ satisfies the condition
$(x\alpha\,|\,y\beta)=\bar{\alpha}(x\,|\,y)\beta$ for $x,y\in\mathbb{H}^{2n}$
and $\alpha,\beta\in\mathbb{H}$.)

The group $\mathbb{G}=\operatorname{SO}^{*}(4n)$ is the group of automorphisms
of $\mathbb{H}^{2n}$ preserving the form $(\,|\,)$, i.e.,

$\mathbb{G}=\left\\{g\in\operatorname{GL}(2n,\mathbb{H})\,\big{|}\,\bar{g}^{t}J_{n}g=J_{n}\right\\}.$

The operations bar and transpose are defined by passing to the complex
matrices: if $X$ is any $2n\times 2n$ quaternionic matrix, we write it as
$X=U+jV$ with $U,V$ complex and identify $X$ with the $4n\times 4n$ complex
matrix $\left(\begin{smallmatrix}U&-\bar{V}\cr
V&\bar{U}\end{smallmatrix}\right)$. Then

$\bar{X}=\overline{\begin{pmatrix}U&-\bar{V}\cr
V&\bar{U}\end{pmatrix}}=\bar{U}+j\bar{V};\ X^{t}=\begin{pmatrix}U&-\bar{V}\cr
V&\bar{U}\end{pmatrix}^{t}=U^{t}-j\bar{V}^{t}.$

Then one has $\overline{XY}=\bar{X}\bar{Y}$ and $(XY)^{t}=Y^{t}X^{t}$. The
reader is cautioned that the operations bar and transpose cannot be performed
directly on the quaternionic matrix in the usual way, but their composition
can, since

$\overline{(U+jV)}^{t}=(\bar{U}+j\bar{V})^{t}=\bar{U}^{t}-jV^{t}=\bar{U}^{t}+\bar{V}^{t}\bar{j}.$

Upon writing $g\in\mathbb{G}$ as $\left(\begin{smallmatrix}A&B\cr
C&D\end{smallmatrix}\right)$ with $n\times n$ (quaternionic) blocks and
writing out the condition $\bar{g}^{t}J_{n}g=J_{n}$, we see

(2.27) $\mathbb{G}=\operatorname{SO}^{*}(4n)=\left\\{\begin{pmatrix}A&B\cr
C&D\end{pmatrix}\,\big{|}\,\bar{A}^{t}C=\bar{C}^{t}A,\,\bar{B}^{t}D=\bar{D}^{t}B,\,\bar{A}^{t}D-\bar{C}^{t}B=I_{n}\right\\}.$

Clearly, $\mathbb{G}$ acts on the skew Hermitian quaternionic Lagrangian
Grassmannian $\operatorname{LGr}^{*}(\mathbb{H}^{2n})$, the manifold of all
Lagrangian subspaces of $\mathbb{H}^{2n}$ with respect to $(\,|\,)$, and this
action is transitive by Witt’s Theorem 1.1. Thus
$\operatorname{LGr}^{*}(\mathbb{H}^{2n})=\mathbb{G}/\mathbb{P}$, where
$\mathbb{P}$ is the stabilizer in $\mathbb{G}$ of the standard Lagrangian
subspace

$L_{0}=\\{(x_{1},\dots,x_{n},0,\dots,0)\,\big{|}\,x_{1},\dots,x_{n}\in\mathbb{H}\\}\subset\mathbb{H}^{2n}.$

It follows that

(2.28) $\mathbb{P}=\left\\{\begin{pmatrix}A&B\cr
0&D\end{pmatrix}\in\operatorname{GL}(2n,\mathbb{H})\,\big{|}\,\bar{B}^{t}D=\bar{D}^{t}B,\,\bar{A}^{t}D=I_{n}\right\\}.$

Let $\sigma$ be an involution of $\mathbb{G}$ given by (1.3), i.e., by
$\sigma(g)=I_{n,n}gI_{n,n}$. Then $\mathbb{G}^{\sigma}$ is equal to the Levi
subgroup $\mathbb{L}$ of $\mathbb{P}$ and it consists of block diagonal
matrices in $\mathbb{G}$, i.e.,

$\mathbb{G}^{\sigma}=\mathbb{L}=\left\\{\begin{pmatrix}A&0\cr
0&(\bar{A}^{t})^{-1}\end{pmatrix}\,\big{|}\,A\in\operatorname{GL}(n,\mathbb{H})\right\\}\cong\operatorname{GL}(n,\mathbb{H})\quad\text{via}\
\begin{pmatrix}A&0\cr 0&(\bar{A}^{t})^{-1}\end{pmatrix}\leftrightarrow A.$

Since $\mathbb{K}$ consists of the fixed points of the Cartan involution
$\theta(g)=(\bar{g}^{t})^{-1}$, and since $g\in\mathbb{G}$ is equivalent to
$(\bar{g}^{t})^{-1}=J_{n}gJ_{n}^{-1}$, we see that $g\in\mathbb{K}$ if and
only if $gJ_{n}=J_{n}g$. This implies

(2.29) $\mathbb{K}=\left\\{\begin{pmatrix}A&-C\cr
C&A\end{pmatrix}\in\operatorname{GL}(2n,\mathbb{H})\,\big{|}\,\bar{A}^{t}C=\bar{C}^{t}A,\,\bar{A}^{t}A+\bar{C}^{t}C=I_{n}\right\\}.$

We now also see that

$\mathbb{P}\cap\mathbb{K}=\mathbb{K}^{\sigma}=\left\\{\begin{pmatrix}A&0\cr
0&A\end{pmatrix}\in\operatorname{GL}(2n,\mathbb{H})\,\big{|}\,\bar{A}^{t}A=I_{n}\right\\}\cong\operatorname{Sp}(n)\quad\text{via}\
\begin{pmatrix}A&0\cr 0&A\end{pmatrix}\leftrightarrow A.$

We claim that $\mathbb{K}\cong\operatorname{U}(2n)$. To see this, we consider
a different copy $\mathbb{G}^{\prime}$ of $\operatorname{SO}^{*}(4n)$ inside
$\operatorname{GL}(2n,\mathbb{H})$, the one preserving the skew Hermitian form

$\langle x,y\rangle=\bar{x}^{t}iy=\sum_{r=1}^{2n}\bar{x}_{r}iy_{r}.$

So $\mathbb{G}^{\prime}$ is the subgroup of $\operatorname{GL}(2n,\mathbb{H})$
consisting of matrices $g$ such that $\bar{g}^{t}ig=iI_{2n}$, and upon writing
$g=U+jV$ with $U,V$ complex, we see

(2.30)
$\mathbb{G}^{\prime}=\left\\{U+jV\in\operatorname{GL}(2n,\mathbb{H})\,\big{|}\,\bar{U}^{t}U+\bar{V}^{t}V=I_{2n},\,U^{t}V=V^{t}U\right\\}.$

Since $g\in\mathbb{G}^{\prime}$ is equivalent to $(\bar{g}^{t})^{-1}=-igi$,
$\theta g=g$ is equivalent to $ig=gi$. Writing $g=U+jV$ with $U,V$ complex, we
see

(2.31)
$\mathbb{K}^{\prime}=\left\\{U+j0\in\operatorname{GL}(2n,\mathbb{H})\,\big{|}\,\bar{U}^{t}U=I_{2n}\right\\}.$

So $\mathbb{K}^{\prime}$ is the usual $\operatorname{U}(2n)$, embedded into
$\operatorname{GL}(2n,\mathbb{H})$ as $U(n)+j0$.

To show an explicit connection between $\mathbb{G}$ and $\mathbb{G}^{\prime}$
and also between $\mathbb{K}$ and $\mathbb{K}^{\prime}$, we note that the
matrix

$T_{n}=\frac{1}{\sqrt{2}}\begin{pmatrix}I_{n}&-iI_{n}\cr
jI_{n}&-kI_{n}\end{pmatrix}$

satisfies $\bar{T}_{n}^{t}iT_{n}=J_{n}$. It follows that

$T_{n}\mathbb{G}T_{n}^{-1}=\mathbb{G}^{\prime},$

and since $\theta T_{n}=T_{n}$, also
$T_{n}\mathbb{K}T_{n}^{-1}=\mathbb{K}^{\prime}=\operatorname{U}(2n)$.
Moreover, $T_{n}(\mathbb{P}\cap\mathbb{K})T_{n}^{-1}$ is the standard
$\operatorname{Sp}(n)$ inside $\operatorname{U}(2n)$, embedded as matrices of
the form $\left(\begin{smallmatrix}U&-\bar{V}\cr
V&\bar{U}\end{smallmatrix}\right)$.

Now Proposition 2.4 implies the following result, which can also be found in
[CN06].

######  Proposition 2.32.

The skew Hermitian quaternionic Lagrangian Grassmannian
$\mathbb{G}/\mathbb{P}=\operatorname{LGr}^{*}(\mathbb{H}^{2n})$ is
diffeomorphic to $\operatorname{U}(2n)/\operatorname{Sp}(n)$.

######  Corollary 2.33.

The cohomology ring (with complex coefficients) of the skew Hermitian
quaternionic Lagrangian Grassmannian
$\operatorname{LGr}^{*}(\mathbb{H}^{2n})\cong\operatorname{U}(2n)/\operatorname{Sp}(n)$
is described by Theorem 4.31.

Finally, we describe the group $G_{\mathbb{R}}=\mathbb{G}^{\sigma\theta}$.
Since $\sigma\theta(g)=I_{n,n}(\bar{g}^{t})^{-1}I_{n,n}$ and since
$g\in\mathbb{G}$ is equivalent to $(\bar{g}^{t})^{-1}=J_{n}gJ_{n}^{-1}$,

$\sigma\theta(g)=I_{n,n}J_{n}gJ_{n}^{-1}I_{n,n}=D_{n}gD_{n},\qquad
g\in\mathbb{G}.$

Thus $\sigma\theta(g)=g$ is equivalent to $gD_{n}=D_{n}g$. It follows that

$\mathbb{G}^{\sigma\theta}=\left\\{g=\begin{pmatrix}A&B\cr
B&A\end{pmatrix}\,\big{|}\,\bar{A}^{t}B=\bar{B}^{t}A,\,\bar{A}^{t}A-\bar{B}^{t}B=I_{n}\right\\}.$

Conjugating $\left(\begin{smallmatrix}A&B\cr B&A\end{smallmatrix}\right)$ by
$\left(\begin{smallmatrix}I_{n}&I_{n}\cr I_{n}&-I_{n}\end{smallmatrix}\right)$
we get the matrix $\left(\begin{smallmatrix}A+B&0\cr
0&A-B\end{smallmatrix}\right)$, and the conditions
$\bar{A}^{t}B=\bar{B}^{t}A,\,\bar{A}^{t}A-\bar{B}^{t}B=I_{n}$ imply
$\overline{(A+B)^{t}}(A-B)=I_{n}$, so $A-B=(\overline{(A+B)^{t}})^{-1}$.
Conversely, starting from the matrix $\left(\begin{smallmatrix}Z&0\cr
0&(\bar{Z}^{t})^{-1}\end{smallmatrix}\right)$ and setting
$A=\frac{1}{2}(Z+(\bar{Z}^{t})^{-1})$, $B=\frac{1}{2}(Z-(\bar{Z}^{t})^{-1})$,
we get $\bar{A}^{t}B=\bar{B}^{t}A,\,\bar{A}^{t}A-\bar{B}^{t}B=I_{n}$. Thus

$\mathbb{G}^{\sigma\theta}\cong\left\\{\begin{pmatrix}Z&0\cr
0&(\bar{Z}^{t})^{-1}\end{pmatrix}\,\big{|}\,Z\in\operatorname{GL}(n,\mathbb{H})\right\\}\cong\operatorname{GL}(n,\mathbb{H})\quad\text{via}\
\begin{pmatrix}Z&0\cr 0&(\bar{Z}^{t})^{-1}\end{pmatrix}\leftrightarrow Z.$

###  2.7. The quadric cases

In this subsection $\mathbb{F}$ is equal to $\mathbb{R}$ or $\mathbb{C}$. The
following is taken from [Tho16, Chapter 4.4]

Let $f$ be a nondegenerate symmetric bilinear form on $\mathbb{F}^{n+2}$. The
quadric $Q_{f}(\mathbb{F})$ is defined to be the subset of the projective
space $P^{n+1}(\mathbb{F})$:

$Q_{f}(\mathbb{F})=\\{x=(x_{1}:\ldots:x_{n+2})\in
P^{n+1}(\mathbb{F})\>|\>f(x,x)=0\\}.$

If $\mathbb{F}=\mathbb{R}$ then $f$ has normal form with matrix $I_{p+1,q+1}$
(if the form is definite, $Q_{f}(\mathbb{R})$ does not contain projective
lines) and we denote $Q_{f}(\mathbb{R})$ by $Q_{p,q}(\mathbb{R})$. If
$\mathbb{F}=\mathbb{C}$ then all $f$ are equivalent and we denote
$Q_{f}(\mathbb{C})$ by $Q_{n}(\mathbb{C})$. The group
$\mathbb{G}=\operatorname{SO}(p+1,q+1)_{e}$ acts transitively on the quadric
$Q_{p,q}(\mathbb{R})$ with parabolic
$\mathbb{P}=\mathrm{Stab}(1:0:\ldots:1:0:\ldots:0)$. The maximal compact
subgroup of $\mathbb{G}$ is
$G=\mathbb{K}=\operatorname{SO}(p+1)\times\operatorname{SO}(q+1)$ and
$K=\mathbb{K}\cap\mathbb{P}=S(\operatorname{O}(p)\times\operatorname{O}(q))$.
In this setting $Q_{p,q}(\mathbb{R})$ is equal to the symmetric space

$G/K=\operatorname{SO}(p+1)\times\operatorname{SO}(q+1)/S(\operatorname{O}(p)\times\operatorname{O}(q)),\qquad
p+q>2.$

The symmetric space is not an irreducible symmetric space, it has a double
cover by spheres $S^{p}\times S^{q}$, however it is indecomposable. When
$\mathbb{F}=\mathbb{C}$ then $\operatorname{SO}_{n+2}(\mathbb{C})$ acts
transitively on $Q_{n+2}(\mathbb{C})$ and the parabolic
$\mathbb{P}=\mathrm{Stab}(1:i:\ldots:0)$ has abelian unipotent radical,
furthermore

$Q_{n}(\mathbb{C})=\operatorname{SO}_{n+2}(\mathbb{C})/\mathrm{Stab}(1:i:\ldots:0)\simeq\operatorname{SO}(n+2)/\operatorname{SO}(n)\times\operatorname{SO}(2)\quad
n\geq 3.$

As a compact symmetric space, $Q_{n}(\mathbb{C})$ coincides with
$\operatorname{SO}(n+2)/\operatorname{SO}(n)\times\operatorname{SO}(2)$, which
is equal to a double cover of the Grassmannian of $2$-planes in
$\mathbb{R}^{n+2}$. We leave the calculation of the cohomology of double
covers of Grassmannians and these quadrics to future work.

##  3\. The structure of $C(\mathfrak{p})^{K}$ and
$(\textstyle{\bigwedge}\mathfrak{p})^{K}$

###  3.1. The decomposition of the spin module

Let $G/K$ be a compact symmetric space corresponding to an involution $\sigma$
of $G$ and let $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$ be the
decomposition of the complexified Lie algebra $\mathfrak{g}$ of $G$ into
eigenspaces of $\sigma$. In particular, $\mathfrak{k}$ is the complexified Lie
algebra of $K$. We note that if $G_{\mathbb{R}}/K$ is the noncompact dual of
$G/K$, then $\sigma$ corresponds to the Cartan involution of $G_{\mathbb{R}}$
($\sigma$ coincides with $\theta$ on $G_{\mathbb{R}}$).

We define the Clifford algebra $C(\mathfrak{p})$ using the nondegenerate
invariant symmetric bilinear form $B$ on $\mathfrak{g}$ obtained by extending
the Killing form over the center of $\mathfrak{g}$. Alternatively, for matrix
groups from our table, we can replace $B$ by the trace form
$\operatorname{tr}XY$. Recall that $C(\mathfrak{p})$ is the associative unital
algebra generated by $\mathfrak{p}$, with relations $XY+YX=2B(X,Y)$,
$X,Y\in\mathfrak{p}$.

Let $S$ be a spin module for $C(\mathfrak{p})$. Recall that $S$ is constructed
as follows. Let $\mathfrak{p}^{+}$ and $\mathfrak{p}^{-}$ be two maximal
isotropic subspaces of $\mathfrak{p}$, dual under $B$. Let
$S=\textstyle{\bigwedge}\mathfrak{p}^{+}$, with elements of
$\mathfrak{p}^{+}\subset C(\mathfrak{p})$ acting by wedging, and elements of
$\mathfrak{p}^{-}\subset C(\mathfrak{p})$ acting by contracting. If
$\dim\mathfrak{p}$ is even,
$\mathfrak{p}=\mathfrak{p}^{+}\oplus\mathfrak{p}^{-}$ and hence this
determines $S$ completely. Moreover, $S$ is the only irreducible
$C(\mathfrak{p})$-module, and $C(\mathfrak{p})=\operatorname{End}S$. If
$\dim\mathfrak{p}$ is odd, then
$\mathfrak{p}=\mathfrak{p}^{+}\oplus\mathfrak{p}^{-}\oplus\mathbb{C}Z$ where
$Z$ is an element of $\mathfrak{p}$ not contained in
$\mathfrak{p}^{+}\oplus\mathfrak{p}^{-}$, such that $B(Z,Z)=1$. Now we can
make $Z$ act on $\textstyle{\bigwedge}\mathfrak{p}^{+}$ in two different ways;
it can act by $1$ on
$\textstyle{\bigwedge}^{\operatorname{even}}\mathfrak{p}^{+}$ and by $-1$ on
$\textstyle{\bigwedge}^{\operatorname{odd}}\mathfrak{p}^{+}$, or by $-1$ on
$\textstyle{\bigwedge}^{\operatorname{even}}\mathfrak{p}^{+}$ and by $1$ on
$\textstyle{\bigwedge}^{\operatorname{odd}}\mathfrak{p}^{+}$. In this way we
get two inequivalent $C(\mathfrak{p})$-modules $S_{1}$ and $S_{2}$. These are
the only irreducible $C(\mathfrak{p})$-modules and
$C(\mathfrak{p})=\operatorname{End}S_{1}\oplus\operatorname{End}S_{2}$. In the
following, $S$ denotes either one of these two modules. For more details about
Clifford algebras, spin modules, and also pin and spin groups, see [HP06,
Ch.2].

Since the pin group $\operatorname{Pin}(\mathfrak{p})$ is contained in
$C(\mathfrak{p})$, the pin double cover $\widetilde{K}$ of $K$ acts on $S$.
Recall that $\widetilde{K}$ is obtained from the following pullback diagram

$\begin{CD}\widetilde{K}@>{}>{}>\operatorname{Pin}(\mathfrak{p})\\\
@V{}V{}V@V{}V{}V\\\ K@>{}>{}>\operatorname{O}(\mathfrak{p})\end{CD}$

where the map $K\to\operatorname{O}(\mathfrak{p})$ is given by the adjoint
action of $K$ on $\mathfrak{p}$.

It now follows that

(3.1)
$C(\mathfrak{p})^{K}=C(\mathfrak{p})^{\widetilde{K}}=\left\\{\begin{matrix}\operatorname{End}_{\widetilde{K}}S,\qquad\qquad\quad\
\quad\dim\mathfrak{p}\text{
even}\cr\operatorname{End}_{\widetilde{K}}S_{1}\oplus\operatorname{End}_{\widetilde{K}}S_{2},\quad\dim\mathfrak{p}\text{
odd}.\end{matrix}\right.$

Since the algebra $C(\mathfrak{p})^{K}$ and its graded version
$(\textstyle{\bigwedge}\mathfrak{p})^{K}$ are of our primary interest, we are
led to study the $\widetilde{K}$-decomposition of $S$. We first study the
decomposition of $S$ under the complexified Lie algebra $\mathfrak{k}$ of
$\widetilde{K}$. This Lie algebra acts on $S$ through the map
$\alpha:\mathfrak{k}\to C(\mathfrak{p})$, which is defined as the action map
$\mathfrak{k}\to\mathfrak{so}(\mathfrak{p})$ followed by the Chevalley map
(i.e., the skew symmetrization)
$\mathfrak{so}(\mathfrak{p})\cong\textstyle{\bigwedge}^{2}\mathfrak{p}\hookrightarrow
C(\mathfrak{p})$. Explicitly, if $b_{i}$ is a basis of $\mathfrak{p}$ with
dual basis $d_{i}$ with respect to the form $B$, then

(3.2) $\alpha(X)=\frac{1}{4}\sum_{i}[X,b_{i}]d_{i},\quad X\in\mathfrak{k}.$

(See [HP06, §2.3.3]; the difference in sign comes from using different
conventions to define the Clifford algebra.)

Let $\mathfrak{t}_{0}$ be a Cartan subalgebra of the (real) Lie algebra
$\mathfrak{k}_{0}$ of $K$ and let
$\mathfrak{t}=(\mathfrak{t}_{0})_{\mathbb{C}}$. Let
$\Delta^{+}(\mathfrak{g},\mathfrak{t})\supseteq\Delta^{+}(\mathfrak{k},\mathfrak{t})$
be compatible choices of positive roots for $(\mathfrak{g},\mathfrak{t})$
respectively $(\mathfrak{k},\mathfrak{t})$. Let $\rho$ respectively
$\rho_{\mathfrak{k}}$ be the corresponding half sums of positive roots. Let
$W_{\mathfrak{g}}$ respectively $W_{\mathfrak{k}}$ be the Weyl groups of
$\Delta(\mathfrak{g},\mathfrak{t})$ respectively
$\Delta(\mathfrak{k},\mathfrak{t})$.

Let $W^{1}_{\mathfrak{g},\mathfrak{k}}$ be the set of minimal length
representatives of $W_{\mathfrak{k}}$-cosets in $W_{\mathfrak{g}}$.
Alternatively,

$W^{1}_{\mathfrak{g},\mathfrak{k}}=\\{\sigma\in
W_{\mathfrak{g}}\,\big{|}\,\sigma\rho\text{ is $\mathfrak{k}$-dominant}\\}.$

It is well known ([Par72]; see also [HP06]) that the decomposition of $S$
under the action of $\mathfrak{k}$ is given by

(3.3) $S=m\cdot\bigoplus_{\sigma\in
W^{1}_{\mathfrak{g},\mathfrak{k}}}E_{\sigma},$

where $E_{\sigma}$ denotes the irreducible finite-dimensional
$\mathfrak{k}$-module with highest weight $\sigma\rho-\rho_{\mathfrak{k}}$,
and the multiplicity $m$ is equal to
$2^{[\frac{1}{2}\tiny{\dim}\mathfrak{a}]}$ where $\mathfrak{a}$ is the
centralizer of $\mathfrak{t}$ in $\mathfrak{p}$ (so that
$\mathfrak{h}=\mathfrak{t}\oplus\mathfrak{a}$ is a Cartan subalgebra of
$\mathfrak{g}$).

Since $m$ is exactly the dimension of the spin module for the Clifford algebra
$C(\mathfrak{a})$, (3.1) and (3.3) imply that

(3.4) $C(\mathfrak{p})^{\mathfrak{k}}\cong
C(\mathfrak{a})\otimes\operatorname{Pr}(S),$

where $\operatorname{Pr}(S)$ is the algebra spanned by the
$\mathfrak{k}$-equivariant projections $\operatorname{pr}_{\sigma}:S\to m\cdot
E_{\sigma}$, $\sigma\in W^{1}_{\mathfrak{g},\mathfrak{k}}$.

If $K$ is connected, then the adjoint action map maps $K$ into
$\operatorname{SO}(\mathfrak{p})$, so $\widetilde{K}$ is the spin double cover
of $K$,

$\begin{CD}\widetilde{K}@>{}>{}>\operatorname{Spin}(\mathfrak{p})\\\
@V{}V{}V@V{}V{}V\\\ K@>{}>{}>\operatorname{SO}(\mathfrak{p}).\end{CD}$

If the double covering map $\widetilde{K}\to K$ does not split, then
$\widetilde{K}$ is connected and (3.3) gives a decomposition of $S$ with
respect to $\widetilde{K}$. If the covering $\widetilde{K}\to K$ splits, then
$\widetilde{K}=K\times\mathbb{Z}_{2}$, where the generator $z$ of
$\mathbb{Z}_{2}$ maps to $1\in K$ under the covering map. This implies that
$z$ maps to the preimage in $\operatorname{Spin}(\mathfrak{p})$ of
$1\in\operatorname{SO}(\mathfrak{p})$, that is to $\pm 1\in C(\mathfrak{p})$.
Thus $z$ acts by the scalar $1$ or $-1$ on $S$, in particular it preserves the
decomposition (3.3), and hence this decomposition is also a decomposition of
the $\widetilde{K}$-module $S$ into irreducibles. To conclude:

######  Proposition 3.5.

If $K$ is connected, then the $\widetilde{K}$-decomposition of $S$ into
irreducibles is the same as the $\mathfrak{k}$-decomposition (3.3).

In general, the $\mathfrak{k}$-decomposition is the same as the decomposition
under $\widetilde{K}_{e}$, the connected component of the identity in
$\widetilde{K}$, but the $\widetilde{K}$-action may combine several
irreducible $\mathfrak{k}$-modules into one irreducible
$\widetilde{K}$-module. More precisely, the component group
$\widetilde{K}/\widetilde{K}_{e}\cong K/K_{e}$ acts by permuting the
components $E_{\sigma}$ of $S$, and the $E_{\sigma}$ combining to produce an
irreducible $\widetilde{K}$-module belong to the same orbit of $K/K_{e}$.
(Recall that $K_{e}$ is a normal subgroup of $K$ and that $K/K_{e}$ is a
finite group.) We will treat the case of disconnected $K$ in Subsections 3.7
and 3.8 below. Before that we describe the structure of
$C(\mathfrak{p})^{\mathfrak{k}}$ more precisely.

###  3.2. Top degree element and Poincaré duality

We identify $C(\mathfrak{p})$ and $\textstyle{\bigwedge}\mathfrak{p}$ using
the Chevalley map, and think of them as one vector space with two different
multiplications.

######  Proposition 3.6.

Let $T$ be the unique (up to a scalar multiple) element of the top wedge of
$\mathfrak{p}$; let $d=\dim\mathfrak{p}$ denote the degree of $T$.

(1) $T$ squares to a nonzero constant with respect to Clifford multiplication;
consequently we can rescale $T$ and assume that $T^{2}=1$.

(2) If $d$ is odd, $T$ is in the center of $C(\mathfrak{p})$. If $d$ is even,
$T$ commutes with $C(\mathfrak{p})_{\operatorname{even}}$.

(3) $T$ is $\mathfrak{k}$-invariant (with respect to the adjoint action).

(4) Clifford multiplication by $T$ from the left is a linear isomorphism from
$\textstyle{\bigwedge}^{j}\mathfrak{p}$ to
$\textstyle{\bigwedge}^{d-j}\mathfrak{p}$, for any $j=0,1,\dots,d$. This
isomorphism, denoted by $*$, preserves the $\mathfrak{k}$-invariants and
therefore gives an isomorphism from
$(\textstyle{\bigwedge}^{j}\mathfrak{p})^{\mathfrak{k}}$ to
$(\textstyle{\bigwedge}^{d-j}\mathfrak{p})^{\mathfrak{k}}$ for any $j$.

(5) The isomorphism $*$ can up to sign be expressed as $x\mapsto\iota_{x}T$.

(6) For any $x,y\in\textstyle{\bigwedge}^{j}\mathfrak{p}$,

$x\wedge*y=B(x,y)T.$

In other words, $*$ is the usual Hodge star operator.

###### Proof.

Let $Z_{1},\dots,Z_{d}$ be an orthonormal basis of $\mathfrak{p}$. Then, up to
a nonzero scalar, $T=Z_{1}\cdots Z_{d}$. It follows that $T^{2}$ is a nonzero
constant, since $Z_{1}\cdots Z_{d}$ squares to $\pm 1$. This proves (1). (2)
is a straightforward computation: one checks that $Z_{1}\cdots Z_{d}$ commutes
with all $Z_{j}$ if $d$ is odd, and with all $Z_{j}Z_{k}$ if $d$ is even.

To prove (3), we note that the adjoint action of $X\in\mathfrak{k}$ on
$C(\mathfrak{p})$ is the same as the Clifford commutator with $\alpha(X)$.
Since $\alpha$ maps $\mathfrak{k}$ into
$C(\mathfrak{p})_{\operatorname{even}}$, the claim follows from (2).

To prove (4), we note that if we set $Z_{I}=Z_{i_{1}}\dots Z_{i_{a}}$ for
$I=\\{i_{1},\dots,i_{a}\\}\subseteq\\{1,\dots,d\\},$ then $TZ_{I}=\pm
Z_{I^{c}},$ where $I^{c}=\\{1,\dots,d\\}\setminus I$. This implies (4). (5)
follows from the fact that Clifford multiplication by $y\in\mathfrak{p}$
equals $\iota_{y}+{\varepsilon}_{y}$ where ${\varepsilon}_{y}$ denotes wedging
by $y$. Since $T$ is of top degree, it is annihilated by all
${\varepsilon}_{y}$, $y\in\mathfrak{p}$, and this implies the claim.

To prove (6), we note that both sides of the equation are bilinear in $x$ and
$y$, so we can assume $x=Z_{I}$, $y=Z_{J}$ for some
$I,J\subseteq\\{1,\dots,d\\}$. If $I\neq J$, both sides of the equation are
zero. Finally, if $I=J$, then we are to check that $Z_{I}\wedge*Z_{I}=T$,
which is a straightforward computation. ∎

######  Lemma 3.7.

Let $x$ be any element of $C(\mathfrak{p})$ such that $x^{2}=1$ (with respect
to Clifford multiplication). Then $B(x,x)=1$, where $B$ is the extended
Killing form on $C(\mathfrak{p})\cong\textstyle{\bigwedge}\mathfrak{p}$.
Consequently the elements $1$ and $x$ span a subalgebra of $C(\mathfrak{p})$
isomorphic to the Clifford algebra on the one-dimensional space $\mathbb{C}x$.

In particular, if $T$ is the top element of $C(\mathfrak{p})$ as above,
rescaled so that $T^{2}=1$, then $B(T,T)=1$ and
$\operatorname{span}_{\mathbb{C}}(1,T)$ is a subalgebra of $C(\mathfrak{p})$
is isomorphic to the Clifford algebra $C(\mathbb{C}T)$.

###### Proof.

This follows from the fact that the constant term of $x^{2}$ is $\iota_{x}x$,
so

$1=B(1,1)=B(\iota_{x}x,1)=B(x,x\wedge 1)=B(x,x).$

∎

###  3.3. $C(\mathfrak{p})^{\mathfrak{k}}$ and
$(\textstyle{\bigwedge}\mathfrak{p})^{\mathfrak{k}}$ in the equal rank cases

The equal rank cases on our list are:

$\displaystyle
G/K=\operatorname{U}(p+q)/\operatorname{U}(p)\times\operatorname{U}(q)\quad$
$\displaystyle\text{ (Subsection \ref{real gras})};$ $\displaystyle
G/K=\operatorname{Sp}(p+q)/\operatorname{Sp}(p)\times\operatorname{Sp}(q)\quad$
$\displaystyle\text{ (Subsection \ref{real gras})};$ $\displaystyle
G/K=\operatorname{SO}(2p+2q)/\mathrm{S}(\operatorname{O}(2p)\times\operatorname{O}(2q))\quad$
$\displaystyle\text{ (Subsection \ref{real gras})};$ $\displaystyle
G/K=\operatorname{SO}(2p+2q+1)/\mathrm{S}(\operatorname{O}(2p)\times\operatorname{O}(2q+1))\quad$
$\displaystyle\text{ (Subsection \ref{real gras})};$ $\displaystyle
G/K=\operatorname{Sp}(n)/\operatorname{U}(n)\quad$ $\displaystyle\text{
(Subsection \ref{real lagr grass})};$ $\displaystyle
G/K=\operatorname{SO}(2n)/\operatorname{U}(n)\quad$ $\displaystyle\text{
(Subsection \ref{real orth lagr grass})}.$

In each of these cases the situation is as in Kostant’s email (see the
introduction). In other words, the spin module $S$ is multiplicity free under
$\mathfrak{k}$ and since $\dim\mathfrak{p}$ is even,
$C(\mathfrak{p})=\operatorname{End}S$. Therefore Schur’s lemma implies that
$C(\mathfrak{p})^{\mathfrak{k}}=\operatorname{End}_{\mathfrak{k}}S=\operatorname{Pr}(S)$,
the algebra spanned by the $\mathfrak{k}$-equivariant projections to the
$\mathfrak{k}$-irreducible constituents of the spin module. The map
$\alpha:\mathfrak{k}\to C(\mathfrak{p})$ from (3.2) extends to
$U(\mathfrak{k})$ and its restriction to the center $Z(\mathfrak{k})$ of
$U(\mathfrak{k})$ is the algebra homomorphism

$\alpha_{\mathfrak{k}}:Z(\mathfrak{k})\to C(\mathfrak{p})^{\mathfrak{k}}.$

(The notation $\alpha_{\mathfrak{k}}$ is to distinguish this map from the
analogous map $\alpha_{K}$ on the level of $K$-invariants; for connected $K$,
there is no difference between these two maps.)

Since the $\mathfrak{k}$-infinitesimal character of $E_{\sigma}$ corresponds
to $\sigma\rho$ under the Harish-Chandra isomorphism
$Z(\mathfrak{k})\cong\mathbb{C}[\mathfrak{t}^{*}]^{W_{\mathfrak{k}}}$, we can
identify $\alpha_{\mathfrak{k}}$ with

(3.8)
$\alpha_{\mathfrak{k}}:\mathbb{C}[\mathfrak{t}^{*}]^{W_{\mathfrak{k}}}\to
C(\mathfrak{p})^{\mathfrak{k}},\qquad\alpha_{\mathfrak{k}}(P)=\sum_{\sigma\in
W^{1}_{\mathfrak{g},\mathfrak{k}}}P(\sigma\rho)\operatorname{pr}_{\sigma},$

where $\operatorname{pr}_{\sigma}$ denotes the $\mathfrak{k}$-equivariant
projection $S\to E_{\sigma}$.

######  Proposition 3.9.

The map $\alpha_{\mathfrak{k}}$ of (3.8) is a filtered algebra homomorphism,
which doubles the degree. Here the filtration on the algebra
$\mathbb{C}[\mathfrak{t}^{*}]^{W_{\mathfrak{k}}}$ is induced by the grading,
while the filtration on the algebra $C(\mathfrak{p})^{\mathfrak{k}}$ is
inherited from $C(\mathfrak{p})$.

###### Proof.

The claim follows from the fact that $\alpha_{\mathfrak{k}}$ is the
restriction of $\alpha:U(\mathfrak{k})\to C(\mathfrak{p})$ given by extending
(3.2). ∎

In the next subsection we consider a more general setting. We will in
particular prove that the map (3.8) is onto; consequently, the map
$\operatorname{gr}\alpha_{\mathfrak{k}}:\mathbb{C}[\mathfrak{t}^{*}]^{W_{\mathfrak{k}}}\to(\textstyle{\bigwedge}\mathfrak{p})^{\mathfrak{k}}$
is also onto. Moreover, we will give a description of $\ker\alpha$ and of
$\ker\operatorname{gr}\alpha=\operatorname{gr}\ker\alpha$. It is clear from
(3.8) that $\ker\alpha_{\mathfrak{k}}$ consists of polynomials vanishing at
all $\sigma\rho$, $\sigma\in W^{1}_{\mathfrak{g},\mathfrak{k}}$. Thus
$\ker\alpha$ contains all $W_{\mathfrak{g}}$-invariant polynomials on
$\mathfrak{t}^{*}$ that vanish at $\rho$ (and thus automatically on all
$\sigma\rho$); we will prove that these polynomials in fact generate
$\ker\alpha_{\mathfrak{k}}$. Likewise, we will see that
$\ker\operatorname{gr}\alpha$ is generated by $W_{\mathfrak{g}}$-invariant
polynomials on $\mathfrak{t}^{*}$ vanishing at $0$.

###  3.4. Relative coinvariant algebra and filtered deformations

Let $W$ be a finite group inside $\operatorname{GL}({\mathfrak{t}})$, with
subgroup $H\subset W$. Let $\nu\in{\mathfrak{t}}^{*}$ be a point such that
$\mathrm{Stab}_{W}(\nu)=\\{\operatorname{id}\\}$. Let $\mathbb{C}[W]$ denote
the algebra of functions from $W$ to $\mathbb{C}$ with pointwise
multiplication. Give $\mathbb{C}[W]$ basis $\\{f_{w}:w\in W\\}$,
$f_{w}(w^{\prime})=\delta_{w,w^{\prime}}1$.

######  Definition 3.10.

Define $\operatorname{Ev}_{\nu}:\mathbb{C}[\mathfrak{t}^{*}]\to\mathbb{C}[W]$
by

$\operatorname{Ev}_{\nu}(p)=\sum_{w\in W}p(w\nu)f_{w}.$

Restricting $\operatorname{Ev}_{\nu}$ to $\mathbb{C}[\mathfrak{t}^{*}]^{H}$ we
define
$\operatorname{Ev}_{\nu}^{H}:\mathbb{C}[\mathfrak{t}^{*}]^{H}\to\mathbb{C}[W]^{H}=\mathbb{C}[W/H]$.

######  Lemma 3.11.

The map $\operatorname{Ev}_{\nu}$ is a surjective $W$-module and algebra
homomorphism and $\operatorname{Ev}_{\nu}^{H}$ is a surjective algebra
homomorphism.

###### Proof.

Clearly $\operatorname{Ev}_{\nu}$ is a $W$ module homomorphism. Let $p_{\nu}$
be a linear polynomial that is zero on $\nu$ and non-zero on all $w\nu$, for
all $w\neq 1.$ The polynomial $\prod_{w\in W\setminus\\{1\\}}wp_{\nu}$
evaluated on $\nu$ is non-zero and is zero on every other element in the orbit
of $\nu$. Suitably scaled, $\operatorname{Ev}_{\nu}(\prod_{w\in
W\setminus\\{1\\}}wp_{\nu})=f_{1}.$ Since $f_{1}$ is a cyclic generator for
the module $\mathbb{C}[W]$ and is in the image of $\operatorname{Ev}_{\nu}$
then the homomorphism is surjective. Since
$f_{w}f_{w^{\prime}}=\delta_{ww^{\prime}}f_{w}$, then
$\operatorname{Ev}_{\nu}(p)\operatorname{Ev}_{\nu}(q)=\sum_{w\in
W}p(w\nu)f_{w}\sum_{w^{\prime}\in W}q(w^{\prime}\nu)f_{w^{\prime}}$ is equal
to $\sum_{w\in W}pq(w\nu)f_{w}=\operatorname{Ev}_{\nu}(pq)$. Taking $H$
invariants on both sides proves that $\operatorname{Ev}_{\nu}^{H}$ is a
surjective algebra homomorphism. ∎

We define two ideals of $\mathbb{C}[\mathfrak{t}^{*}]$, $I_{W,+}$ is the two
sided graded ideal generated by

$\\{p\in\mathbb{C}[\mathfrak{t}^{*}]^{W}:\deg
p>0\\}=\\{p\in\mathbb{C}[\mathfrak{t}^{*}]^{W}:p(0)=0\\}$

and $I_{W,\nu}$ is the filtered ideal generated by
$\\{p\in\mathbb{C}[\mathfrak{t}^{*}]^{W}:p(\nu)=0\\}$.

######  Lemma 3.12.

The ideal $I_{W,+}$ is equal to $\operatorname{gr}I_{W,\nu}$.

###### Proof.

$I_{W,+}$ is generated by
$\mathbb{C}[\mathfrak{t}^{*}]^{W}_{+}=\\{p\in\mathbb{C}[\mathfrak{t}^{*}]^{W}:p(0)=0\\}$
and $I_{W,\nu}$ is generated by
$\mathbb{C}[\mathfrak{t}^{*}]^{W}_{\nu}=\\{p\in\mathbb{C}[\mathfrak{t}^{*}]^{W}:p(\nu)=0\\}$.
Both of which are codimension $1$ in $\mathbb{C}[\mathfrak{t}^{*}]^{W}$.
Furthermore, $\operatorname{gr}(\mathbb{C}[\mathfrak{t}^{*}]^{W}_{\nu})$ is a
codimension one graded ideal of $\mathbb{C}[\mathfrak{t}^{*}]^{W}$. Since
$\mathbb{C}[\mathfrak{t}^{*}]^{W}_{+}$ is the only graded ideal of codimension
one, then
$\operatorname{gr}\mathbb{C}[\mathfrak{t}^{*}]^{W}_{\nu}=\mathbb{C}[\mathfrak{t}^{*}]^{W}_{+}$.
∎

######  Lemma 3.13.

The kernel of $\operatorname{Ev}_{\nu}$ is equal to $I_{W,\nu}$ and the
$\ker\operatorname{Ev}_{\nu}^{H}=I_{W,\nu}^{H}$.

###### Proof.

The kernel of $\operatorname{Ev}_{\nu}$ is precisely polynomials that evaluate
to zero on the full $W$-orbit of $\nu$. The kernel $I_{W,\nu}$ is generated by
$W$-invariant polynomials that evaluate to zero on $\nu$. Since they are
$W$-invariant they also evaluate to zero on the full $W$-orbit. Hence
$\operatorname{ker}\operatorname{Ev}_{\nu}\supset I_{W,\nu}$. The quotient of
$\mathbb{C}[\mathfrak{t}^{*}]$ by $I_{W,+}$ is the coinvariant algebra, which,
in particular, is of dimension $|W|$. Lemma 3.12 then shows that the
codimension of $I_{W,\nu}$ is $|W|$. Since $\operatorname{Ev}_{\nu}$ is
surjective onto $\mathbb{C}[W]$ then $\ker\operatorname{Ev}_{\nu}$ is also of
codimension $|W|$ and hence
$\operatorname{ker}\operatorname{Ev}_{\nu}=I_{W,\nu}$. The second statement
follows by taking $H$ invariants of both sides. ∎

The polynomials $\mathbb{C}[\mathfrak{t}^{*}]$ (resp.
$\mathbb{C}[\mathfrak{t}^{*}]^{H}$) are graded, therefore the map
$\operatorname{Ev}_{\nu}$ gives $\mathbb{C}[W]$ (resp. $\mathbb{C}[W/H]$) a
filtration.

######  Theorem 3.14.

Let $W\subset\operatorname{GL}({\mathfrak{t}})$ be such that
$\mathbb{C}[\mathfrak{t}^{*}]$ is a free $\mathbb{C}[\mathfrak{t}^{*}]^{W}$
module of rank $|W|$ and let $H$ be any subgroup of $W$. With the filtration
on $\mathbb{C}[W]$ endowed by $\operatorname{Ev}_{\nu}$, the associated graded
algebra of $\mathbb{C}[W]$ is the coinvariant algebra
$\mathbb{C}[\mathfrak{t}^{*}]/\langle\mathbb{C}[\mathfrak{t}^{*}]^{W}_{+}\rangle$,
similarly
$\operatorname{gr}(\mathbb{C}[W/H])\cong\mathbb{C}[\mathfrak{t}^{*}]^{H}/\langle\mathbb{C}[\mathfrak{t}^{*}]^{W}_{+}\rangle_{\mathbb{C}[\mathfrak{t}^{*}]^{H}}.$

###### Proof.

Lemmas 3.12 and 3.13 prove that
$\operatorname{gr}\ker\operatorname{Ev}_{\nu}=I_{W,+}$ and
$\operatorname{gr}(\ker\operatorname{Ev}_{\nu}^{H})=I_{W,+}^{H}$. Since
$\mathbb{C}[W]=\mathbb{C}[\mathfrak{t}^{*}]/\ker(\operatorname{Ev}_{\nu})$
then
$\operatorname{gr}\mathbb{C}[W]=\mathbb{C}[\mathfrak{t}^{*}]/\operatorname{gr}(\ker(\operatorname{Ev}_{\nu}))=\mathbb{C}[\mathfrak{t}^{*}]/I_{W,+}$
which is the definition of the coinvariant algebra of $W$ acting on
$\mathbb{C}[\mathfrak{t}^{*}]$. An identical statement holds for
$\operatorname{Ev}_{\nu}^{H}$. ∎

Hence for any finite group $W$ acting by complex reflections on
${\mathfrak{t}}$ with any subgroup $H$ we can define a filtration on
$\mathbb{C}[W/H]$ such that the associated graded algebra is isomorphic to the
relative coinvariant algebra of $W$ and $H$.

######  Corollary 3.15.

With notation of Subsections 3.1 and 3.3, the map $\alpha_{\mathfrak{k}}$ is
surjective onto $C(\mathfrak{p})^{\mathfrak{k}}$ and the kernel of
$\alpha_{\mathfrak{k}}$ is generated by the $W_{\mathfrak{g}}$ invariant
polynomials in $\mathbb{C}[\mathfrak{t}^{*}]^{W_{\mathfrak{k}}}$ which
evaluate to zero at $\rho$, $(I_{W,\rho})^{W_{\mathfrak{k}}}$. Furthermore,
the map $\operatorname{gr}\alpha_{\mathfrak{k}}$ is surjective onto
$(\textstyle{\bigwedge}\mathfrak{p})^{\mathfrak{k}}$ and the kernel of
$\operatorname{gr}\alpha_{\mathfrak{k}}$ is
$I_{W_{\mathfrak{g}},+}^{W_{\mathfrak{k}}}$.

###### Proof.

The map $\alpha_{\mathfrak{k}}$ defined by (3.8) is given by evaluation of
polynomials in $\mathbb{C}[\mathfrak{t}^{*}]^{W_{\mathfrak{k}}}$ at
$\sigma\rho$ for $\sigma\in W^{1}_{\mathfrak{g},\mathfrak{k}}$, defining an
isomorphism between $C(\mathfrak{p})^{\mathfrak{k}}=\operatorname{Pr}(S)$ and
$\mathbb{C}[W_{G}/W_{K}]$

where $\operatorname{pr}_{\sigma}$ maps to $\sum_{w\in\sigma
W_{\mathfrak{k}}}f_{w}$. So the map $\alpha_{\mathfrak{k}}$ is equal to the
restriction of
$\operatorname{Ev}_{\rho}:\mathbb{C}[\mathfrak{t}^{*}]\to\mathbb{C}[W_{\mathfrak{g}}]$
to $W_{\mathfrak{k}}$ invariants

$\operatorname{Ev}_{\rho}^{W_{\mathfrak{k}}}:\mathbb{C}[\mathfrak{t}^{*}]^{W_{\mathfrak{k}}}\to\mathbb{C}[W_{\mathfrak{g}}/W_{\mathfrak{k}}]=\mathbb{C}[W^{1}_{\mathfrak{g},\mathfrak{k}}].$

Lemma 3.11 then states that $\alpha_{\mathfrak{k}}$ is surjective onto
$C(\mathfrak{p})^{\mathfrak{k}}=\operatorname{Pr}(S)$ and Lemma 3.13 describes
the kernel. Theorem 3.14 provides the statement for
$\operatorname{gr}\alpha_{\mathfrak{k}}$. ∎

###  3.5. $C(\mathfrak{p})^{\mathfrak{k}}$ and
$(\textstyle{\bigwedge}\mathfrak{p})^{\mathfrak{k}}$ in the almost equal rank
case:
$G/K=\operatorname{SO}(2p+2q+2)/S(\operatorname{O}(2p+1)\times\operatorname{O}(2q+1))$

We call this case “almost equal rank”, because $\dim\mathfrak{a}=1$ for all
$p$ and $q$. To see that indeed $\dim\mathfrak{a}=1$, and also for later
purposes, we first describe a Cartan subalgebra
$\mathfrak{h}=\mathfrak{t}\oplus\mathfrak{a}$ of $\mathfrak{g}$.

For the Cartan subalgebra $\mathfrak{t}$ of $\mathfrak{k}$ we choose block
diagonal matrices with diagonal blocks

(3.16) $t_{1}J,\dots,t_{p}J,0,t_{p+1}J,\dots,t_{p+q}J,0,$

where $J=J_{1}=\left(\begin{smallmatrix}0&1\cr-1&0\end{smallmatrix}\right)$,
and $t_{1},\dots,t_{p+q}$ are (complex) scalars. The centralizer
$\mathfrak{a}$ of $\mathfrak{t}$ in $\mathfrak{p}$ is one-dimensional, spanned
by $E_{k\,k+m}-E_{k+m\,k}$. We identify $\mathfrak{t}$ with
$\mathbb{R}^{p+q}\times 0\subset\mathbb{R}^{p+q+1}$ by sending the matrix
(3.16) to $(t_{1},\dots,t_{p+q},0)$, and we identify $\mathfrak{a}$ with
$0\times\mathbb{R}\subset\mathbb{R}^{p+q+1}$ by sending
$E_{k\,k+m}-E_{k+m\,k}$ to $(0,\dots,0,1)$.

Since $\dim\mathfrak{p}$ is odd,
$C(\mathfrak{p})=\operatorname{End}S_{1}\oplus\operatorname{End}S_{2}$, where
$S_{1}$ and $S_{2}$ are the two spin modules. These spin modules are not
isomorphic as $C(\mathfrak{p})$-modules, but they are isomorphic as modules
over $C(\mathfrak{p})_{\operatorname{even}}$, in particular they are
isomorphic as $\mathfrak{k}$-modules. Moreover, the $\mathfrak{k}$-module
$S=S_{1}=S_{2}$ is multiplicity free (since the multiplicity
$m=2^{[\frac{1}{2}\tiny{\dim}\mathfrak{a}]}=1$).

To understand the decomposition
$C(\mathfrak{p})=\operatorname{End}S_{1}\oplus\operatorname{End}S_{2}$ more
explicitly, we first note that the top element $T$ of $C(\mathfrak{p})$, which
is central in $C(\mathfrak{p})$ since $\dim\mathfrak{p}$ is odd, acts as 1 on
$S_{1}$ and as $-1$ on $S_{2}$. Therefore the central idempotents

$\operatorname{pr}_{1}=\frac{1}{2}(1+T),\qquad\operatorname{pr}_{2}=\frac{1}{2}(1-T)$

satisfy the following: $\operatorname{pr}_{1}$ is 1 on $S_{1}$ and 0 on
$S_{2}$, while $\operatorname{pr}_{2}$ is 0 on $S_{1}$ and 1 on $S_{2}$. It
follows that

$\operatorname{End}S_{1}=C(\mathfrak{p})\operatorname{pr}_{1}\qquad\text{ and
}\qquad\operatorname{End}S_{2}=C(\mathfrak{p})\operatorname{pr}_{2}.$

By Proposition 3.6, multiplication by $T$ is an isomorphism between
$C(\mathfrak{p})_{\operatorname{even}}$ and
$C(\mathfrak{p})_{\operatorname{odd}}$, and moreover

(3.17) $C(\mathfrak{p})\cong C(\mathbb{C}T)\otimes
C(\mathfrak{p})_{\operatorname{even}},$

with the isomorphism implemented by the multiplication. It follows that

$\operatorname{End}S_{1}=C(\mathfrak{p})_{\operatorname{even}}\operatorname{pr}_{1}\qquad\text{
and
}\qquad\operatorname{End}S_{2}=C(\mathfrak{p})_{\operatorname{even}}\operatorname{pr}_{2}.$

Namely, $\operatorname{End}S_{i}$ corresponds to $\operatorname{pr}_{i}\otimes
C(\mathfrak{p})_{\operatorname{even}}$ under the decomposition (3.17).

Since the $\mathfrak{k}$-action on $C(\mathbb{C}T)$ is trivial, and since
$\operatorname{End}S_{1}=\operatorname{End}S_{2}=\operatorname{End}S$ as
$\mathfrak{k}$-modules, we see that for any $c\in C(\mathbb{C}T)$, $c\otimes
C(\mathfrak{p})_{\operatorname{even}}$ is a copy of $\operatorname{End}S$. In
particular, $1\otimes
C(\mathfrak{p})_{\operatorname{even}}=C(\mathfrak{p})_{\operatorname{even}}$
is isomorphic to $\operatorname{End}S$, and in the following when we write
$\operatorname{End}S$ we mean this particular copy. It follows that
$\operatorname{End}_{\mathfrak{k}}S=C(\mathfrak{p})_{\operatorname{even}}^{\mathfrak{k}}$;
this is also the image of the map $\alpha_{\mathfrak{k}}$ of (3.8), which now
sends $\mathbb{C}[\mathfrak{t}^{*}]^{W_{\mathfrak{k}}}$ onto
$\operatorname{Pr}(S)=\operatorname{End}_{\mathfrak{k}}S\subset
C(\mathfrak{p})^{\mathfrak{k}}$. ($\operatorname{End}_{\mathfrak{k}}S$ is
equal to the algebra $\operatorname{Pr}(S)$ of projections onto isotypic
components since $S$ is multiplicity free.) Furthermore, an analogue of
Proposition 3.9 holds, with $C(\mathfrak{p})^{\mathfrak{k}}$ replaced by
$\operatorname{Pr}(S)$. Finally, the above discussion shows that

$C(\mathfrak{p})^{\mathfrak{k}}=C(\mathbb{C}T)\otimes\operatorname{Pr}(S).$

We now go back to our Cartan subalgebra
$\mathfrak{h}=\mathfrak{t}\oplus\mathfrak{a}$. Since
$(\mathfrak{g},\mathfrak{t})$-roots are the restrictions of
$(\mathfrak{g},\mathfrak{h})$-roots to $\mathfrak{t}$, we see that
$\Delta(\mathfrak{g},\mathfrak{t})$ is of type $B_{p+q}$, while
$\Delta(\mathfrak{k},\mathfrak{t})$ is of type $B_{p}\times B_{q}$. (On the
other hand, $\Delta(\mathfrak{g},\mathfrak{h})$ is of type $D_{p+q+1}$.)

######  Lemma 3.18.

The filtered algebra $\operatorname{Pr}(S)$ is isomorphic to the filtered
algebra
$C(\mathfrak{p}^{\prime})^{\mathfrak{k}^{\prime}}=\operatorname{Pr}(S^{\prime})$
for the equal rank symmetric space

$G^{\prime}/K^{\prime}=\operatorname{Sp}(p+q)/\operatorname{Sp}(p)\times\operatorname{Sp}(q).$

This algebra is isomorphic to the algebra
$\mathbb{C}[\mathfrak{t}^{*}]^{W_{\mathfrak{k}}^{\prime}}$ modulo the ideal
generated by $\mathbb{C}[\mathfrak{t}^{*}]^{W_{\mathfrak{g}}^{\prime}}$. (We
identify the isomorphic spaces $\mathfrak{t}$ and $\mathfrak{t}^{\prime}$.) It
can be identified with the space $\mathbb{C}[r_{1},\dots,r_{p}]_{\leq q}$,
spanned by monomials of degree at most $q$ in the elementary symmetric
functions $r_{1},\dots,r_{p}$ of the squares of the variables
$x_{1},\dots,x_{p}$, as in Subsection 4.3 below. The degrees of these
monomials as functions of the $x_{i}$ range from 0 to $4pq$ and are divisible
by 4.

###### Proof.

It will be shown in Subsection 4.3 that
$C(\mathfrak{p}^{\prime})^{\mathfrak{k}^{\prime}}$ for $G^{\prime}/K^{\prime}$
is $\mathbb{C}[r_{1},\dots,r_{p}]_{\leq q}$ as in the statement of the lemma.
Since types $B$ and $C$ have the same Weyl group, $W_{\mathfrak{g}}$ and
$W_{\mathfrak{k}}$ are the same for $G/K$ and $G^{\prime}/K^{\prime}$.
Moreover, $\rho=\rho^{\prime}=(p+q,p+q-1,\dots,1)\in\mathfrak{t}^{*}$. Since
the algebra $\operatorname{Pr}(S)$ is isomorphic to the algebra
$\mathbb{C}[\mathfrak{t}^{*}]^{W_{\mathfrak{k}}}$ modulo the ideal generated
by $\mathbb{C}[\mathfrak{t}^{*}]^{W_{\mathfrak{g}}}_{\rho}$, this implies the
lemma. ∎

The degree of the top element $T$ of $C(\mathfrak{p})$ is
$d=\dim\mathfrak{p}=4pq+2p+2q+1$. The algebra $\operatorname{Pr}(S)$ contains
a unique element $t$ of degree $4pq$. Let $e=Tt$ be the corresponding odd
element. Then $e$ is the unique element of lowest odd degree; this degree is

$\deg e=d-4pq=2p+2q+1.$

######  Lemma 3.19.

The elements $t$ and $e$ square to nonzero constants in $C(\mathfrak{p})$.
Therefore, we can rescale these two elements and assume that $t^{2}=e^{2}=1$.

###### Proof.

By Lemma 3.18, the filtered algebra $\operatorname{Pr}(S)$ is isomorphic to
the filtered algebra $C(\mathfrak{p}^{\prime})^{\mathfrak{k}^{\prime}}$ for
$(\mathfrak{g}^{\prime},\mathfrak{k}^{\prime})=(\mathfrak{sp}(p,q),\mathfrak{sp}(p)\times\mathfrak{sp}(q))$.
By Proposition 3.6, the top degree element of
$C(\mathfrak{p}^{\prime})^{\mathfrak{k}^{\prime}}$ squares to a nonzero
constant. So $t^{2}$ is a nonzero constant.

Since $T^{2}=1$, it follows that $e=Tt$ squares to the same constant. ∎

######  Corollary 3.20.

The element $e\in C(\mathfrak{p})^{\mathfrak{k}}$ satisfies $B(e,e)=1$ (here
$B$ is the extended Killing form on
$C(\mathfrak{p})\cong\textstyle{\bigwedge}\mathfrak{p}$). Consequently the
elements $1$ and $e$ span a subalgebra of $C(\mathfrak{p})^{\mathfrak{k}}$
isomorphic to the Clifford algebra on the one-dimensional space $\mathbb{C}e$.
The same is true if we replace $e$ by $t$.

###### Proof.

This follows from Lemma 3.19 and Lemma 3.7. ∎

######  Theorem 3.21.

There are tensor product decompositions

$\displaystyle C(\mathfrak{p})^{\mathfrak{k}}\cong
C(\mathbb{C}e)\otimes\operatorname{Pr}(S);$
$\displaystyle(\textstyle{\bigwedge}\mathfrak{p})^{\mathfrak{k}}\cong\textstyle{\bigwedge}\mathbb{C}e\otimes\operatorname{gr}\operatorname{Pr}(S),$

with the isomorphisms implemented by the multiplication.

###### Proof.

To show that $C(\mathfrak{p})^{\mathfrak{k}}\cong
C(\mathbb{C}e)\otimes\operatorname{Pr}(S)$, we first note that by (3.17),
$\operatorname{Pr}(S)=C(\mathfrak{p})_{\operatorname{even}}$ is a subalgebra
of $C(\mathfrak{p})^{\mathfrak{k}}$ of half the dimension. Moreover, by
Corollary 3.20, $\operatorname{span}\\{1,e\\}$ is a subalgebra of
$C(\mathfrak{p})^{\mathfrak{k}}$ isomorphic to the Clifford algebra
$C(\mathbb{C}e)$ of the space $\mathbb{C}e$. It is thus enough to show that
(Clifford) multiplication by $e$ is injective on $\operatorname{Pr}(S)$. This
follows immediately from $e^{2}=1$. Since $\operatorname{Pr}(S)$ commutes with
$C(\mathfrak{p})^{\mathfrak{k}}$, this concludes the proof of
$C(\mathfrak{p})^{\mathfrak{k}}\cong
C(\mathbb{C}e)\otimes\operatorname{Pr}(S)$.

To prove
$(\textstyle{\bigwedge}\mathfrak{p})^{\mathfrak{k}}\cong\textstyle{\bigwedge}\mathbb{C}e\otimes\operatorname{gr}\operatorname{Pr}(S)$,
we again start from the fact that $\operatorname{gr}\operatorname{Pr}(S)$ is a
subalgebra of $(\textstyle{\bigwedge}\mathfrak{p})^{\mathfrak{k}}$ of half the
dimension. Moreover, $e\wedge e=0$, since the degree of $e\wedge e$ is
$4p+4q+2$, which is even but not divisible by 4. It thus suffices to see that
wedging by $e$ is injective on $\operatorname{gr}\operatorname{Pr}(S)$.

We first note that $e\wedge t=T$ up to (nonzero) scalar. Indeed, since the
degrees match, it is enough to see that $e\wedge t\neq 0$. But

$B(e\wedge t,T)=-B(e,\iota_{t}T)=-B(e,e)\neq 0.$

(We have already seen that $B(e,e)=1$. Alternatively, since $e$ is the only
$\mathfrak{k}$-invariant in its degree, and since $B$ is nondegenerate on
$\mathfrak{k}$-invariants, $B(e,e)\neq 0$.)

Assume now that $p\in\operatorname{gr}\operatorname{Pr}(S)$ is nonzero; we
want to show that $e\wedge p\neq 0$. We can assume $p$ is homogeneous. We will
be done if we can show that there is
$p^{\prime}\in\operatorname{gr}\operatorname{Pr}(S)$ such that $p\wedge
p^{\prime}=t$; then

$(e\wedge p)\wedge p^{\prime}=e\wedge t\neq 0,$

so also $e\wedge p\neq 0$.

By Lemma 3.18, the algebra $\operatorname{Pr}(S)$ is isomorphic (as a filtered
algebra) to the algebra $C(\mathfrak{p}^{\prime})^{\mathfrak{k}^{\prime}}$,
where
$(\mathfrak{g}^{\prime},\mathfrak{k}^{\prime})=(\mathfrak{sp}(p+q),\mathfrak{sp}(p)\times\mathfrak{sp}(q))$.
The corresponding graded algebras are thus also isomorphic. The claim now
follows from Proposition 3.6.(6). ∎

###  3.6. $(\textstyle{\bigwedge}\mathfrak{p})^{K}$ in the primary case and
almost primary case

In this section we cite results from [GHV76] that describe the structure of
$(\textstyle{\bigwedge}\mathfrak{p})^{\mathfrak{k}}$, extend this description
to $(\textstyle{\bigwedge}\mathfrak{p})^{K}$ and explicitly give a generating
subspace when $G/K$ is primary or almost primary.

######  Definition 3.22.

Let
$\lambda_{\mathfrak{k}}=\mathrm{gr}\alpha_{\mathfrak{k}}:U(\mathfrak{k})^{\mathfrak{k}}\to(\textstyle{\bigwedge}\mathfrak{p})^{\mathfrak{k}}$
and let $\lambda_{K}$ be the restriction of $\lambda_{\mathfrak{k}}$ to
$U(\mathfrak{k})^{K}$.

######  Theorem 3.23.

[GHV76, X.4 Th VII] Let $(\mathfrak{g},\mathfrak{k})$ be a symmetric pair of
Lie algebras, with Cartan involution $\theta$. Let
$\mathcal{P}_{\mathfrak{g}}$ be a graded subspace that generates
$\textstyle{\bigwedge}(\mathfrak{g})^{\mathfrak{g}}$ and define the Samelson
subspace $\mathcal{P}_{\mathfrak{a}}=\cal P_{\mathfrak{g}}^{-\theta}$ then
there is an isomorphism of graded algebras

$(\textstyle{\bigwedge}\mathfrak{p})^{\mathfrak{k}}\cong\textstyle{\bigwedge}\cal
P_{\mathfrak{a}}\otimes\operatorname{im}\lambda_{\mathfrak{k}}.$

We extend the graded algebra description of
$(\textstyle{\bigwedge}\mathfrak{p})^{\mathfrak{k}}$ from [GHV76] to
$(\textstyle{\bigwedge}\mathfrak{p})^{K}$ in the below proposition.

######  Proposition 3.24.

Let $G/K$ be a symmetric space with $G$ connected ($K$ may be disconnected).
Then, with notation as in Theorem 3.23, there is an isomorphism of graded
algebras

$(\textstyle{\bigwedge}\mathfrak{p})^{K}\cong\textstyle{\bigwedge}\cal
P_{\mathfrak{a}}\otimes\operatorname{im}\lambda_{K}.$

###### Proof.

When one identifies the de Rham cohomology $H(G/K_{e})$ of the space $G/K_{e}$
with $\textstyle{\bigwedge}{\mathfrak{p}}^{\mathfrak{k}}$ and the de Rham
cohomology $H(G)$ of $G$ with
$\textstyle{\bigwedge}({\mathfrak{g}})^{\mathfrak{g}}$, then the map on
cohomology from $H(G/K_{e})$ to $H(G)$ is given by the inclusion of
$(\textstyle{\bigwedge}\mathfrak{p})^{\mathfrak{k}}$ into
$\textstyle{\bigwedge}\mathfrak{g}$ followed by the projection of
$\textstyle{\bigwedge}\mathfrak{g}$ onto
$(\textstyle{\bigwedge}\mathfrak{g})^{\mathfrak{g}}$ along
$\mathfrak{g}\cdot\textstyle{\bigwedge}\mathfrak{g}$ (see [GHV76, X.4] for
extra details). Both of these maps are $K$-module homomophisms and the image
of the composition is $\textstyle{\bigwedge}\cal
P_{\mathfrak{a}}\subset\textstyle{\bigwedge}\cal P_{\mathfrak{g}}$ [GHV76, X.4
Th VII (2)]. Since $(\textstyle{\bigwedge}\mathfrak{g})^{\mathfrak{g}}$ is
$G$-fixed (and hence $K$-fixed) and the above map is a $K$-module
homomorphism, we can conclude that the subspace of
$(\textstyle{\bigwedge}{\mathfrak{p}})^{\mathfrak{k}}$ congruent to
$\textstyle{\bigwedge}\cal P_{\mathfrak{a}}$ is $K$-fixed. The space
$\textstyle{\bigwedge}({\mathfrak{p}})^{K}$ is equal to

$(\textstyle{\bigwedge}({\mathfrak{p}})^{\mathfrak{k}})^{K}\cong(\textstyle{\bigwedge}(\cal
P_{\mathfrak{a}})\otimes\operatorname{im}\lambda_{\mathfrak{k}})^{K}=\textstyle{\bigwedge}(\cal
P_{\mathfrak{a}})\otimes(\operatorname{im}\lambda_{\mathfrak{k}})^{K},$

the second equality following from the fact that the first tensorand is
entirely $K$-fixed. To finish, note that
$(\operatorname{im}\lambda_{\mathfrak{k}})^{K}=\operatorname{im}\lambda_{K}$,
hence

$(\textstyle{\bigwedge}{\mathfrak{p}})^{K}\cong\textstyle{\bigwedge}\cal
P_{\mathfrak{a}}\otimes\operatorname{im}\lambda_{K}.$

∎

######  Definition 3.25.

The symmetric space $G/K$ is primary if
$W_{\mathfrak{g},\mathfrak{t}}=W_{\mathfrak{k}}$ and almost primary if
$W_{\mathfrak{g},\mathfrak{t}}=W_{K}\neq W_{\mathfrak{k}}$.

######  Definition 3.26.

Define $\cal P_{\wedge}(\mathfrak{p})$ to be the subspace of
$(\textstyle{\bigwedge}\mathfrak{p})^{K}$ orthogonal to square of the
augmentation ideal $((\textstyle{\bigwedge}\mathfrak{p})^{K}_{+})^{2}$.

######  Proposition 3.27.

[Oni94, proposition 4 p 105] Suppose that the algebra $A$ is isomorphic to an
exterior algebra, and let $S$ be a subset of $A$. Then the following are
equivalent:

  1. (a)

the algebra $A$ is generated by $S$ and $1$;

  2. (b)

the augmentation ideal $A_{+}$ is equal to the $A$-submodule generated by $S$;

  3. (c)

the augmentation ideal $A_{+}$ is equal to
$\mathrm{span}_{\mathbb{C}}(S)\oplus A_{+}^{2}$.

In particular, Proposition 3.27 shows that modifying a generating set $S$ by
any elements from $A_{+}^{2}$ retains the property of generating $A$; we will
use this to show that $P_{\wedge}(\mathfrak{p})$ generates
$(\textstyle{\bigwedge}\mathfrak{p})^{K}$ in the (almost) primary case.

######  Corollary 3.28.

Suppose that $A$ is a graded algebra with non-degenerate bilinear form such
that $A$ is isomorphic to an exterior algebra, and different graded components
are orthogonal to each other. Let $R$ be the subspace of $A_{+}$ orthogonal to
$A_{+}^{2}$ then $A$ is generated by $R$.

###### Proof.

Since $A$ is isomorphic to an exterior algebra, suppose $P$ is any graded
subspace of $A$ such that $A=\textstyle{\bigwedge}P$. The proof follows by
performing a Gram-Schmidt algorithm on $P$ modifying by elements in
$A_{+}^{2}$ until the generating subspace is orthogonal to $A_{+}^{2}$. By
Proposition 3.27, at each step of the Gram-Schmidt process the new subspace
still generates $A$ and is graded since the different graded components are
orthogonal. The end result is a graded subspace $P^{\prime}$ orthogonal to
$A_{+}^{2}$ that generates $A$. Hence we have a direct sum of orthogonal
components

$A=\mathbb{C}\oplus P^{\prime}\oplus A_{+}^{2},$

and $P^{\prime}$ is contained in $R$. Any element in $R\setminus P^{\prime}$
would be orthogonal to $P^{\prime}$, $\mathbb{C}$ and $A_{+}^{2}$ thus
contradicting the fact that the form on $A$ is non-degenerate, hence
$R=P^{\prime}$ and $R$ generates $A$. ∎

######  Theorem 3.29.

Let $G/K$ be primary or almost primary, then the inclusion $\cal
P_{\wedge}(\mathfrak{p})\hookrightarrow(\textstyle{\bigwedge}\mathfrak{p})^{K}$
extends to an isomorphism of graded algebra

$(\textstyle{\bigwedge}\mathfrak{p})^{K}=\textstyle{\bigwedge}\cal
P_{\wedge}(\mathfrak{p}).$

###### Proof.

If $G/K$ is primary then $\operatorname{im}\lambda_{\mathfrak{k}}$ is
$\mathbb{C}$ and
$(\textstyle{\bigwedge}\mathfrak{p})^{\mathfrak{k}}\cong\textstyle{\bigwedge}\cal
P_{\mathfrak{a}}$, if $G/K$ is almost primary then
$\operatorname{im}\lambda_{K}$ is $\mathbb{C}$ and
$(\textstyle{\bigwedge}\mathfrak{p})^{K}\cong\textstyle{\bigwedge}\cal
P_{\mathfrak{a}}$. Hence, in both cases
$(\textstyle{\bigwedge}\mathfrak{p})^{K}$ is isomorphic to an exterior
algebra, denote this isomorphism by $f:\textstyle{\bigwedge}\cal
P_{\mathfrak{a}}\cong(\textstyle{\bigwedge}\mathfrak{p})^{K}$. Then
$(\textstyle{\bigwedge}\mathfrak{p})^{K}$ is generated by the graded subspace
$f(\cal P_{\mathfrak{a}})$ and the form on
$(\textstyle{\bigwedge}\mathfrak{p})^{K}$ induced by the Killing form on
$\mathfrak{p}$ is non degenerate on $(\textstyle{\bigwedge}\mathfrak{p})^{K}$
with differing graded components orthogonal. Hence Corollary 3.28 proves that
$P_{\wedge}(\mathfrak{p})$ (Definition 3.26) generates
$(\textstyle{\bigwedge}\mathfrak{p})^{K}$;
$(\textstyle{\bigwedge}\mathfrak{p})^{K}=\textstyle{\bigwedge}\cal
P_{\wedge}(\mathfrak{p})$. ∎

The degrees of $\cal P_{\wedge}(\mathfrak{p})$ are the same as the degrees of
$\cal P_{\mathfrak{a}}=\cal P_{\mathfrak{g}}^{-\theta}$ which are given in
[GHV76, Table I,II, III p. 492-496] and repeated below for reference. There is
paper in preparation [CGKP] that will prove a transgression theorem when $G/K$
is primary or almost primary and will directly give the degrees of $\cal
P_{\wedge}(\mathfrak{p})$.

Table 2. Degrees of $\cal P_{\wedge}({\mathfrak{p}})$

$G$ | $K$ | degrees of $\cal P_{\wedge}({\mathfrak{p}})$
---|---|---
Group |  |
$\operatorname{U}_{2n+1}(\mathbb{R})^{2}$ | $\operatorname{U}_{2n+1}(\mathbb{R})$ | $4p-1:1\leq p\leq n$
$\operatorname{U}_{2n}(\mathbb{R})^{2}$ | $\operatorname{U}_{2n}(\mathbb{R})$ |  $4p-1:1\leq p\leq n-1$ and $2n-1$
$\operatorname{U}_{n}(\mathbb{C})^{2}$ | $\operatorname{U}_{n}(\mathbb{C})$ | $2p-1:1\leq p\leq n$
$\operatorname{U}_{n}(\mathbb{H})^{2}$ | $\operatorname{U}_{n}(\mathbb{H})$ | $4p-1:1\leq p\leq n$
Primary |  |
$\operatorname{U}_{2n+1}(\mathbb{C})$ | $\operatorname{U}_{2n+1}(\mathbb{R})$ | $4p-3:1\leq p\leq n+1$
$\operatorname{U}_{2n}(\mathbb{C})$ | $\operatorname{U}_{n}(\mathbb{H})$ | $4p-3:1\leq p\leq n$
Almost Primary |  |
$\operatorname{U}_{2n}(\mathbb{C})$ | $\operatorname{U}_{2n}(\mathbb{R})$ | $4p-3:1\leq p\leq n$

(Recall $\operatorname{U}_{n}(\mathbb{R})=\operatorname{O}(n)$,
$\operatorname{U}_{n}(\mathbb{C})=\operatorname{U}(n)$, and
$\operatorname{U}_{n}(\mathbb{H})=\operatorname{Sp}(n)$.)

###  3.7. $C(\mathfrak{p})^{K}$ and $(\textstyle{\bigwedge}\mathfrak{p})^{K}$
for disconnected $K$: the case
$G/K=\operatorname{SO}(k+m)/S(\operatorname{O}(k)\times\operatorname{O}(m))$

To understand what happens with the decomposition (3.3) when $K$ is
disconnected, we consider the group theoretic Weyl group $W_{K}$ defined as

(3.30) $W_{K}=N_{K}(\mathfrak{t})/Z_{K}(\mathfrak{t}),$

where $N_{K}(\mathfrak{t})$ denotes the normalizer in $K$ of the Cartan
subalgebra $\mathfrak{t}$ of $\mathfrak{k}$, while $Z_{K}(\mathfrak{t})$
denotes the centralizer of $\mathfrak{t}$ in $K$. It is well known (see
[Kna96, Theorem 4.54]) that for connected $K$, $W_{K}=W_{\mathfrak{k}}$, the
Weyl group of the root system of $(\mathfrak{k},\mathfrak{t})$. Analogously,
we define $W_{G}$, which is equal to
$W_{\mathfrak{g}}=W_{\mathfrak{g},\mathfrak{t}}$ since $G$ is assumed
connected.

We note that it looks like we should consider the groups $W_{\widetilde{K}}$
and $W_{\widetilde{K}_{e}}$, but these are in fact the same as $W_{K}$
respectively $W_{K_{e}}$. This follows from the fact that the adjoint action
of $k\in\widetilde{K}$ is the same as the adjoint action of $\pi(k)$ where
$\pi$ denotes the covering map from $\widetilde{K}$ to $K$.

Let now
$G/K=\operatorname{SO}(k+m)/S(\operatorname{O}(k)\times\operatorname{O}(m))$.
In these cases $K$ is disconnected and the group theoretic Weyl group $W_{K}$
may be different from $W_{\mathfrak{k}}$. In the following we describe $W_{K}$
explicitly.

The group $K=S(\operatorname{O}(k)\times\operatorname{O}(m))$ has two
connected components, and we choose the following explicit representative $s$
of the disconnected component:

(3.31) $\displaystyle s$ $\displaystyle=$
$\displaystyle\operatorname{diag}(\underbrace{1,\dots,1,-1}_{k},\underbrace{1,\dots,1,-1}_{m})\quad\text{if
}k\text{ and }m\text{ are both even};$ $\displaystyle s$ $\displaystyle=$
$\displaystyle\operatorname{diag}(\underbrace{1,\dots,1,-1}_{k},\underbrace{-1,\dots,-1}_{m})\quad\text{if
}k\text{ is even and }m\text{ is odd};$ $\displaystyle s$ $\displaystyle=$
$\displaystyle\operatorname{diag}(\underbrace{-1,\dots,-1}_{k},\underbrace{-1,\dots,-1}_{m})\quad\text{if
}k\text{ and }m\text{ are both odd}.$

For the Cartan subalgebra $\mathfrak{t}$ of $\mathfrak{k}$ we choose block
diagonal matrices with diagonal blocks

$\displaystyle t_{1}J,\dots,t_{p}J,t_{p+1}J,\dots,t_{p+q}J$
$\displaystyle\qquad\text{if }(k,m)=(2p,2q);$ $\displaystyle
t_{1}J,\dots,t_{p}J,t_{p+1}J,\dots,t_{p+q}J,0$ $\displaystyle\qquad\text{if
}(k,m)=(2p,2q+1);$ $\displaystyle
t_{1}J,\dots,t_{p}J,0,t_{p+1}J,\dots,t_{p+q}J,0$ $\displaystyle\qquad\text{if
}(k,m)=(2p+1,2q+1);,$

where $J=J_{1}=\left(\begin{smallmatrix}0&1\cr-1&0\end{smallmatrix}\right)$,
and $t_{1},\dots,t_{p+q}$ are (complex) scalars.

In the equal rank cases $(k,m)=(2p,2q)$ and $(k,m)=(2p,2q+1)$, $\mathfrak{t}$
is also a Cartan subalgebra of $\mathfrak{g}$, while for $(k,m)=(2p+1,2q+1)$,
as noted in Subsection 3.5, a Cartan subalgebra for $\mathfrak{g}$ is
$\mathfrak{h}=\mathfrak{t}\oplus\mathfrak{a}$ where $\mathfrak{a}$ is one-
dimensional, spanned by $E_{k\,k+m}-E_{k+m\,k}$. For $(k,m)=(2p,2q)$ and
$(k,m)=(2p,2q+1)$, we identify $\mathfrak{t}$ with $\mathbb{R}^{p+q}$ by
sending the above described matrix to $(t_{1},\dots,t_{p+q})$. For
$(k,m)=(2p+1,2q+1)$ we identify $\mathfrak{t}$ with $\mathbb{R}^{p+q}\times
0\subset\mathbb{R}^{p+q+1}$ by sending the above described matrix to
$(t_{1},\dots,t_{p+q},0)$, and we identify $\mathfrak{a}$ with
$0\times\mathbb{R}\subset\mathbb{R}^{p+q+1}$ by sending
$E_{k\,k+m}-E_{k+m\,k}$ to $(0,\dots,0,1)$. In this last case we see that
$\Delta(\mathfrak{g},\mathfrak{t})$ is of type $B_{p+q}$, while