# question about equivariant embeddings of riemannian symmetric domains

Here by riemannian symmetric domain is understood an riemannian symmetric space with only factors of non-compact types. Such domains are realized as quotients of the form $D=G/K$, where $G$ is a connected semi-simple real linear algebraic group, $K$ a maximal compact subgroup, which is the fixed part of a Cartan involution $\theta$ on $G$.

And by equivariant embeddings is understood a smooth embedding of symmetric domains: $i:D_1\rightarrow D$, with $D_1$ defined by a connected semi-simple subgroup $G_1\subset G$, stabilized by $\theta$. In this case $\theta_{G_1}$ is a Cartan involution on $G_1$, with fixed part $K_1$, and thus $D_1$ is isomorphic to $G_1/K_1$. If one regards $D=G/K$ as the set of conjugates of $\theta$ under $G$, then $D_1$ is the $G_1$-orbit in $G\theta=D$.

The question is to understand refinement of such embeddings, i.e. what kind of chain of equivariant embeddings can one get like $D_1\rightarrow D_2\rightarrow D$?

More specifically, consider the centralizer $Z=Z(G_1,G)$ of $G_1$ in $G$. Note that $Z$ is a connected subgroup. Two cases arise:

(1) $Z$ is non-compact; then one can show that $G_1$ extends to a non-trivial parabolic subgroup of $G$, i.e. $G_1\subset P\subsetneq G$.

(2) $Z$ is compact; then $G_1$ cannot be extended to a non-trivial parabolic, and any subgroup of $G$ containing $G_1$ has to be reductive.

for a reference of the characterization above, see "non-divergence of translates of certain algebraic measures", lemma 5.1, by A.Eskin, S.Mozes, N.Shah, in GAFA 7(1997), pp.48-80

Write $N^\circ=N^\circ(G_1,G)$ for the connected component of the normalizer of $G_1$ in $G$. Then this reductive subgroup is also stable under the Cartan involution $\theta$, and the corresponding symmetric subdomain given as the $N^\circ$-orbit of $\theta$ in $D=G\theta$.

If (2) happens, then $N^\circ$ gives the same symmetric subdomain $D_1$; otherwise in case (1) $N^\circ$ could give a larger symmeric subdomain $D_2$, in which $D_1$ serves as a factor $D_2=D_1\times D_1'$.

I would like to know

(i) how to characterize more geometrically the difference between (1) and (2);

(ii) is (2) transitive? namely, if one has a chain of equivariant embeddings $D_1\rightarrow D_2\rightarrow D$ given by semi-simple subgroups $G_1\subset G_2\subset G$ stable under a common Cartan involution, such that $Z(G_1,G_2)$ and $Z(G_2,G)$ both compact, can one show $Z(G_1,G)$ compact also?

(iii) does any difference occur if one works with equivariant embeddings of Hermitian symmetric domains?

Sorry for the lengthy presentation.

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To construct examples and/or counter-examples, you may look into Satake's papers jstor.org/stable/2373012?origin=crossref and springerlink.com/content/u18811726062q041 –  Mikhail Borovoi Oct 3 '10 at 9:27

(An answer rather than comment, for length issues...) First, yes, as Mikhail B. notes, Satake considered the hermitian case at length, circa 1965, both in some papers in Amer J. Math. (I think), and in a book printed by Princeton and Iwanami-Shoten about 1979. Many examples of the hermitian case were exploited from late 1970s through 1990s in various papers (Shimura, Harris, Kudla, myself) for applications to arithmetical aspects of (holomorphic) automorphic forms on classical domains.

In the context of automorphic forms, many other examples of equivariant imbeddings were considered, from the early 1980s, by Rallis-PiatetskiShapiro, Shimura, Harris, Kudla, myself, and many others, for similar purposes, but often without any special attention to holomorphic things, thus ceasing to be especially concerned with hermitian symmetric spaces.

Jacquet (et al)'s discussion of ("relative") trace formulas attached to subgroups fixed by involutions successfully abstracts some aspects of the harmonic analysis of automorphic forms in such settings.

Schlichtkrull, van den Baan, and their collaborators have considered the "free space" version of the problem at length, in many interesting papers.

The questions about compactness-or-not, and related, are somewhat subtle, so far as I understand from extensive examples. Perhaps cf the Schlichtkrull/vdBaan/et-alia work.

It is true that "compact periods" of automorphic forms are vastly more tractable (and, ironically, less interesting) than non-compact periods.

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