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Let G be a semisimple algebraic group defined over an algebraically closed field. We provide some criteria for normality and rational singularities of G-saturation under certain circumstances. Our results are applied to determine when the commuting variety over simple Lie algebra of low rank is normal and Cohen-Macaulay. We also present some interesting connections between injective modules and normality (resp. rational singularities) of their G-saturations. Finally, we generalize a machinery used to study singularities of nilpotent orbit closures.

In this two-part paper, we introduce a $$p$$ -adic analytic analogue of Backelin and Kremnizer’s construction of the quantum flag variety of a semisimple algebraic group, when $$q$$ is not a root of unity and $$\vert q-1\vert

For [math] a product of Severi–Brauer varieties, we conjecture that if the Chow ring of [math] is generated by Chern classes, then the canonical epimorphism from the Chow ring of [math] to the graded ring associated to the coniveau filtration of the Grothendieck ring of [math] is an isomorphism. We show this conjecture is equivalent to the condition that if [math] is a split semisimple algebraic group of type [math] , [math] is a Borel subgroup of [math] and [math] is a standard generic [math] -torsor, then the canonical epimorphism from the Chow ring of [math] to the graded ring associated with the coniveau filtration of the Grothendieck ring of [math] is an isomorphism. In certain cases we verify this conjecture.

Let $G$ be a simply-connected semisimple algebraic group over an algebraically closed field of characteristic $p$, assumed to be larger than the Coxeter number. The "support variety" of a $G$-module $M$ is a certain closed subvariety of the nilpotent cone of $G$, defined in terms of cohomology for the first Frobenius kernel $G_1$. In the 1990s, Humphreys proposed a conjectural description of the support varieties of tilting modules; this conjecture has been proved for $G = \mathrm{SL}_n$ in earlier work of the second author. In this paper, we show that for any $G$, the support variety of a tilting module always contains the variety predicted by Humphreys, and that they coincide (i.e., the Humphreys conjecture is true) when $p$ is sufficiently large. We also prove variants of these statements involving "relative support varieties.", 54 pages, 2 color figures. v3: minor corrections and additions

Let $G$ be a connected semisimple algebraic group and let $H \subset G$ be a connected reductive subgroup. Given a flag variety $X$ of $G$, a result of Vinberg and Kimelfeld asserts that $H$ acts spherically on $X$ if and only if for every irreducible representation $R$ of $G$ realized in the space of sections of a homogeneous line bundle on $X$ the restriction of $R$ to $H$ is multiplicity free. In this case, the information on restrictions to $H$ of all such irreducible representations of $G$ is encoded in a monoid, which we call the restricted branching monoid. In this paper, we review the cases of spherical actions on flag varieties of simple groups for which the restricted branching monoids are known (this includes the case where $H$ is a Levi subgroup of $G$) and compute the restricted branching monoids for all spherical actions on flag varieties that correspond to triples $(G,H,X)$ satisfying one of the following two conditions: (1) $G$ is simple and $H$ is a symmetric subgroup of $G$; (2) $G = \mathrm{SL}_n$., Comment: v3: 39 pages, some corrections in the list of references

This paper studies the “reduction mod p” method, which constructs large classes of representations for a semisimple algebraic group G from representations for the corresponding Lusztig quantum group U ζ at a p r -th root of unity. The G-modules arising in this way include the Weyl modules, the induced modules, and various reduced versions of these modules. We present a relation between Ext G n ( V , W ) and Ext U ζ n ( V ′ , W ′ ) , when V , W are obtained from V ′ , W ′ by reduction mod p. Since the dimensions of Ext n -spaces for U ζ -modules are known in many cases, our result guarantees the existence of many new extension classes and homomorphisms between rational G-modules. One application is a new proof of James Franklin's result on certain homomorphisms between two Weyl modules. We also provide some examples which show that the p-th root of unity case and a general p r -th root of unity case are essentially different.

Let $\mathfrak{g}=\mathfrak{g}_{\bar{0}}\oplus\mathfrak{g}_{\bar{1}}$ be a finite-dimensional simple Lie superalgebra of type $D(2,1;\alpha)$, $G(3)$ or $F(4)$ over $\mathbb{C}$. Let $G$ be the simply connected semisimple algebraic group over $\mathbb{C}$ such that $\mathrm{Lie}(G)=\mathfrak{g}_{\bar{0}}$. Suppose $e\in\mathfrak{g}_{\bar{0}}$ is nilpotent. We describe the centralizer $\mathfrak{g}^{e}$ of $e$ in $\mathfrak{g}$ and its centre $\mathfrak{z}(\mathfrak{g}^{e})$ especially. We also determine the labelled Dynkin diagram for $e$. We prove theorems relating the dimension of $\left(\mathfrak{z}(\mathfrak{g}^{e})\right)^{G^{e}}$ and the labelled Dynkin diagram., Comment: 31 pages, 14 figures, accepted by Journal of algebra and its applications

Let H be a semisimple algebraic group, K a maximal compact subgroup of $$G:=H({{\mathbb {R}}})$$ G : = H ( R ) , and $$\Gamma \subset H({{\mathbb {Q}}})$$ Γ ⊂ H ( Q ) a congruence arithmetic subgroup. In this paper, we generalize existing subconvex bounds for Hecke–Maass forms on the locally symmetric space $$\Gamma \backslash G/K$$ Γ \ G / K to corresponding bounds on the arithmetic quotient $$\Gamma \backslash G$$ Γ \ G for cocompact lattices using the spectral function of an elliptic operator. The bounds obtained extend known subconvex bounds for automorphic forms to non-trivial K-types, yielding such bounds for new classes of automorphic representations. They constitute subconvex bounds for eigenfunctions on compact manifolds with both positive and negative sectional curvature. We also obtain new subconvex bounds for holomorphic modular forms in the weight aspect.

Let $G$ be a semisimple algebraic group with Lie algebra $\mathfrak g$. For a nilpotent $G$-orbit $\mathcal O\subset\mathfrak g$, let $d_\mathcal O$ denote the maximal dimension of a subspace $V\subset \mathfrak g$ that is contained in the closure of $\mathcal O$. In this note, we prove that $d_\mathcal O \le \frac{1}{2}\dim\mathcal O$ and this upper bound is attained if and only if $\mathcal O$ is a Richardson orbit. Furthermore, if $V$ is $B$-stable and $\dim V= \frac{1}{2}\dim\mathcal O$, then $V$ is the nilradical of a polarisation of $\mathcal O$. Every nilpotent orbit closure has a distinguished $B$-stable subspace constructed via an $\mathfrak{sl}_2$-triple, which is called the Dynkin ideal. We then characterise the nilpotent orbits $\mathcal O$ such that the Dynkin ideal (1) has the minimal dimension among all $B$-stable subspaces $\mathfrak c$ such that $\mathfrak c\cap\mathcal O$ is dense in $\mathfrak c$, or (2) is the only $B$-stable subspace $\mathfrak c$ such that $\mathfrak c\cap\mathcal O$ is dense in $\mathfrak c$., 23 pages, to appear in the "Journal of the Australian Math. Soc."

Horospheres for an action of a semisimple algebraic group G on an affine variety X are the generic orbits of a maximal unipotent subgroup U ⊂ G or, equivalently, the generic fibers of the categorical quotient of the variety X by the action of U, which is defined by the values of the highest weight functions. The remaining fibers of this quotient (which we call degenerate horospheres) for a certain class of spherical G-varieties containing all simply connected symmetric spaces are studied.

In this paper we discuss the existence of certain classes of cuspidal automorphic representations having non-zero Fourier coefficients for a general semisimple algebraic group $ G$ defined over a number field $ k$ such that its Archimedean group $ G_\infty $ is not compact. When $ G$ is quasi-split over $ k$, we obtain a result on existence of generic cuspidal automorphic representations which generalize results of Vignéras, Henniart, and Shahidi. We also discuss: (i) the existence of cuspidal automorphic forms with non-zero Fourier coefficients for congruence of subgroups of $ G_\infty $, and (ii) applications related to the work of Bushnell and Henniart on generalized Whittaker models.

Let X be the variety of Borel subgroups of a split semisimple algebraic group G over a field, twisted by a generic G-torsor. Conjecturally, the canonical epimorphism of the Chow ring $$\mathop {\mathrm {CH}}\nolimits X$$ onto the associated graded ring GK(X) of the topological filtration on the Grothendieck ring K(X) is an isomorphism. We prove the new cases $$G={\text {Spin}}(11)$$ and $$G={\text {Spin}}(12)$$ of this conjecture. On an equivalent note, we compute the Chow ring $$\mathop {\mathrm {CH}}\nolimits Y$$ of the highest orthogonal grassmannian Y for the generic 11- and 12-dimensional quadratic forms of trivial discriminant and Clifford invariant. In particular, we describe the torsion subgroup of the Chow group $$\mathop {\mathrm {CH}}\nolimits Y$$ and determine its order which is equal to $$16\;777\; 216$$ . On the other hand, we show that the Chow group $$\mathop {\mathrm {CH}}\nolimits _0Y$$ of 0-cycles on Y is torsion-free.

The paper concerns a certain subcategory of the category of representations for a semisimple algebraic group $G$ in characteristic $p$, which arise from the semisimple modules for the corresponding quantum group at a $p$-th root of unity. The subcategory, thus, records the cohomological difference between quantum groups and algebraic groups. We define translation functors in this category and use them to obtain information on the irreducible characters for $G$ when the Lusztig character formula does not hold., Comment: accepted for Special Issue on Representation Theory in Sci China Math

We look at the centralizer in a semisimple algebraic group $G$ of a regular nilpotent element, and show that its closure in the wonderful compactification is isomorphic to the Peterson variety. It follows that the closure in the wonderful compactification of the centralizer $G^x$ of any regular element $x$ is isomorphic to the closure of a general $G^x$-orbit in the flag variety. We also give a description of the $G^e$-orbit structure of the Peterson variety., Final version, to appear in Represent. Theory

In this article we investigate ordinary and equivariant Rost motives. We provide an equivariant motivic decomposition of the variety of full flags of a split semisimple algebraic group G over a field, a motivic decomposition of E/B for a G-torsor E over a smooth base scheme, study torsion subgroup of the Chow group of E/B, define some equivariant Rost motives over a field and some ordinary Rost motives over a general base scheme, and relate equivariant Rost motives with classifying spaces of some algebraic groups.

We prove that the variety of complete flags for any semisimple algebraic group is rigid in any smooth family of Fano manifolds.

Let $G$ be a complex simply-laced semisimple algebraic group of rank $r$ and $B$ a Borel subgroup. Let $\mathbf i \in [r]^n$ be a word and let $\mathbf \ell = (\ell_1,\dots,\ell_n)$ be a sequence of non-negative integers. Grossberg and Karshon introduced a virtual lattice polytope associated to $\mathbf i$ and $\mathbf \ell$ called a twisted cube, whose lattice points encode the character of a $B$-representation. More precisely, lattice points in the twisted cube, counted with sign according to a certain density function, yields the character of the generalized Demazure module determined by $\mathbf i$ and $\mathbf \ell$. In recent work, the author and Harada described precisely when the Grossberg-Karshon twisted cube is untwisted, i.e., the twisted cube is a closed convex polytope, in the situation when the integer sequence $\mathbf \ell$ comes from a weight $\lambda$ of $G$. However, not every integer sequence $\mathbf \ell$ comes from a weight of $G$. In the present paper, we interpret untwistedness of Grossberg-Karshon twisted cubes associated to any word $\mathbf i$ and any integer sequence $\mathbf \ell$ using the combinatorics of $\mathbf i$ and $\mathbf \ell$. Indeed, we prove that the Grossberg-Karshon twisted cube is untwisted precisely when $\mathbf i$ is hesitant-jumping-$\mathbf \ell$-walk-avoiding., Comment: Keywords: Grossberg-Karshon twisted cubes, pattern avoidance, character formula, generalized Demazure modules. arXiv admin note: text overlap with arXiv:1407.8543

According to a 2018 preprint by Nobuaki Yagita, the conjecture on a relationship between [Formula: see text]- and Chow theories for a generically twisted flag variety of a split semisimple algebraic group [Formula: see text], due to the author, fails for [Formula: see text] the spinor group [Formula: see text]. Yagita’s tools include a Brown–Peterson version of algebraic cobordism, ordinary and connective Morava [Formula: see text]-theories, as well as Grothendieck motives related to various cohomology theories over fields of characteristic [Formula: see text]. We provide a proof using only the [Formula: see text]- and Chow theories themselves and extend the (slightly modified) example to arbitrary characteristic.

Let G be a simply connected, semisimple algebraic group of type B4 or D4 over an algebraically closed field of characteristic p > 0. We determine the characters of certain simple modules for these groups by calculating the composition factors of the Weyl modules.

A closed subgroup of a semisimple algebraic group is called irreducible if it lies in no proper parabolic subgroup. In this paper we classify all irreducible A 1 subgroups of exceptional algebraic groups G. Consequences are given concerning the representations of such subgroups on various G-modules: for example, the conjugacy classes of irreducible A 1 subgroups are determined by their composition factors on the adjoint module of G.

Let $G$ be a semisimple algebraic group over a field of characteristic $p > 0$. We prove that the dual Weyl modules for $G$ all have $p$-filtrations when $p$ is not too small. Moreover, we give applications of this theorem to $p^n$-filtrations for $n > 1$, to modules containing the Steinberg module as a tensor factor, and to the Donkin conjecture on modules having $p$-filtrations., minor revisions and updates

We introduce a rigid analytification of the quantized coordinate algebra of a semisimple algebraic group and a quantized Arens-Michael envelope of the enveloping algebra of the corresponding Lie algebra, working over a non-archimedean field and when $q$ is not a root of unity. We show that these analytic quantum groups are topological Hopf algebras and Fr\'echet-Stein algebras. We then introduce an analogue of the BGG category $\mathcal{O}$ for the quantum Arens-Michael envelope and show that it is equivalent to the category $\mathcal{O}$ for the corresponding quantized enveloping algebra., Comment: 36 pages, Section 5 and more references added

Let G be a connected, semisimple algebraic group over a field k whose characteristic is very good for G. In a canonical manner, one associates to a nilpotent element X is an element of Lie(G) a parabolic subgroup P - in characteristic zero, P may be described using an sl(2)-triple containing X; in general, P is the "instability parabolic" for X as in geometric invariant theory. In this setting, we are concerned with the center Z(C) of the centralizer C of X in G. Choose a Levi factor L of P, and write d for the dimension of the center Z(L). Finally, assume that the nilpotent element X is even. In this case, we can deform Lie(L) to Lie(C), and our deformation produces a d-dimensional subalgebra of Lie(Z(C)). Since Z(C) is a smooth group scheme, it follows that dim Z(C) >= d = dim Z(L). In fact, Lawther and Testerman have proved that dim Z(C) = dim Z(L). Despite only yielding a partial result, the interest in the method found in the present work is that it avoids the extensive case-checking carried out by Lawther and Testerman.

Let $G$ be a semisimple algebraic group over an algebraically closed field of characteristic $p \geq 0$. At the 1966 International Congress of Mathematicians in Moscow, Robert Steinberg conjectured that two elements $a, a' \in G$ are conjugate in $G$ if and only if $f(a)$ and $f(a')$ are conjugate in $\operatorname{GL}(V)$ for every rational irreducible representation $f: G \rightarrow \operatorname{GL}(V)$. Steinberg showed that the conjecture holds if $a$ and $a'$ are semisimple, and also proved the conjecture when $p = 0$. In this paper, we give a counterexample to Steinberg's conjecture. Specifically, we show that when $p = 2$ and $G$ is simple of type $C_5$, there exist two non-conjugate unipotent elements $u, u' \in G$ such that $f(u)$ and $f(u')$ are conjugate in $\operatorname{GL}(V)$ for every rational irreducible representation $f: G \rightarrow \operatorname{GL}(V)$., Comment: to appear in Transform. Groups

Let $\G$ be a semisimple algebraic group over a number field $K$, $\mathcal{S}$ a finite set of places of $K$, $K_\mathcal{S}$ the direct product of the completions $K_v, v \in \mathcal{S}$, and $\OO$ the ring of $\mathcal{S}$-integers of $K$. Let $G = \G(K_\mathcal{S})$, $\Gamma = \G(\OO)$ and $\pi:G \rightarrow G/\Gamma$ the quotient map. We describe the closures of the locally divergent orbits ${T\pi(g)}$ %in $G/\Gamma$ where $T$ is a maximal $K_\mathcal{S}$-split torus in $G$. If $\# S = 2$ then the closure $\overline{T\pi(g)}$ is a finite union of $T$-orbits stratified in terms of parabolic subgroups of $\G \times \G$ and, consequently, $\overline{T\pi(g)}$ is homogeneous (i.e., $\overline{T\pi(g)}= H\pi(g)$ for a subgroup $H$ of $G$) if and only if ${T\pi(g)}$ is closed. On the other hand, if $\# \mathcal{S} > 2$ and $K$ is not a $\mathrm{CM}$-field then $\overline{T\pi(g)}$ is homogeneous for $\G = \mathbf{SL}_{n}$ and, generally, non-homogeneous but squeezed between closed orbits of two reductive subgroups of equal semisimple $K$-ranks for $\G \neq \mathbf{SL}_{n}$. As an application, we prove that $\overline{f(\OO^{n})} = K_{\mathcal{S}}$ for the class of non-rational locally $K$-decomposable homogeneous forms $f \in K_{\mathcal{S}}[x_1, \cdots, x_{n}]$., Comment: arXiv admin note: text overlap with arXiv:1502.02297

The probabilistic Waring problem for finite simple groups asks whether every word of the form $w_1w_2$, where $w_1$ and $w_2$ are non-trivial words in disjoint sets of variables, induces almost uniform distribution on finite simple groups with respect to the $L^1$ norm. Our first main result provides a positive solution to this problem. We also provide a geometric characterization of words inducing almost uniform distribution on finite simple groups of Lie type of bounded rank, and study related random walks. Our second main result concerns the probabilistic $L^{\infty}$ Waring problem for finite simple groups. We show that for every $l \ge 1$ there exists $N = N(l)$, such that if $w_1, \ldots , w_N$ are non-trivial words of length at most $l$ in pairwise disjoint sets of variables, then their product $w_1 \cdots w_N$ is almost uniform on finite simple groups with respect to the $L^{\infty}$ norm. The dependence of $N$ on $l$ is genuine. This result implies that, for every word $w = w_1 \cdots w_N$ as above, the word map induced by $w$ on a semisimple algebraic group over an arbitrary field is a flat morphism. Applications to representation varieties, subgroup growth, and random generation are also presented., Comment: 45 pages

We study the general problem of equidistribution of expanding translates of an analytic curve by an algebraic diagonal flow on the homogeneous space $G/\Gamma$ of a semisimple algebraic group $G$. We define two families of algebraic subvarieties of the associated partial flag variety $G/P$, which give the obstructions to non-divergence and equidistribution. We apply this to prove that for Lebesgue almost every point on an analytic curve in the space of $m\times n$ real matrices whose image is not contained in any subvariety coming from these two families, the Dirichlet's theorem on simultaneous Diophantine approximation cannot be improved. The proof combines geometric invariant theory, Ratner's theorem on measure rigidity for unipotent flows, and linearization technique., Comment: 33 Pages. Removed Section 7 from the earlier version

In this chapter, we discuss the representation theory of semisimple algebraic groups. We also sketch the construction of finite dimensional irreducible representations of semisimple algebraic groups. We further discuss the geometric realization of finite dimensional irreducible representations of a semisimple algebraic group (over ℂ).

Let G be a complex semisimple algebraic group. In 2006, Belkale-Kumar defined a new product ⊙0 on the cohomology group H*(G/P,C) of any projective G-homogeneous space G/P. Their definition uses the notion of Levi-movability for triples of Schubert varieties in G/P.

Let G be a split semisimple algebraic group over a field k and let X be the flag variety (i.e., the variety of Borel subgroups) of G twisted by a generic G-torsor. We start a systematic study of the conjecture, raised in [5] in form of a question, that the canonical epimorphism of the Chow ring of X onto the associated graded ring of the topological filtration on the Grothendieck ring of X is an isomorphism. Since the topological filtration in this case is known to coincide with the computable gamma filtration, this conjecture indicates a way to compute the Chow ring. We reduce its proof to the case of k=Q. For simply-connected or adjoint G, we reduce the proof to the case of simple G. Finally, we provide a list of types of simple groups for which the conjecture holds. Besides of some classical types considered previously (namely, A, C, and the special orthogonal groups of types B and D), the list contains the exceptional types G2, F4, and simply-connected E6.

We explore the integration of representations from a Lie algebra to its algebraic group in positive characteristic. An integrable module is stable under the twists by group elements. Our aim is to investigate cohomological obstructions for passing from stability to an algebraic group action. As an application, we prove integrability of bricks for a semisimple algebraic group., Comment: Version 2: Some changes in terminology, examples of over-restricted modules are added. Version 3: mistakes in tables are corrected. Version 4: major revision, the paper is split into two parts, the exponential part is published separately (see arXiv:1807.08698). Version 5: minor corrections. Version 6: final, minor corrections

The ground field \(\mathbb{k}\) is algebraically closed and of characteristic zero. Let G be a connected semisimple algebraic group, and T a maximal torus inside a Borel subgroup B.

Let B be a Borel subgroup of a semisimple algebraic group G and let \(\mathfrak m\) be an abelian nilradical in \(\mathfrak b=\mathrm{Lie} (B)\). Using subsets of strongly orthogonal roots in the subset of positive roots corresponding to \(\mathfrak m\), D. Panyushev [1] gives in particular classification of \(B-\)orbits in \(\mathfrak m\) and \({\mathfrak m}^*\) and states general conjectures on the closure and dimensions of the \(B-\)orbits in both \(\mathfrak m\) and \({\mathfrak m}^*\) in terms of involutions of the Weyl group. Using Pyasetskii correspondence between \(B-\)orbits in \(\mathfrak m\) and \({\mathfrak m}^*\) he shows the equivalence of these two conjectures. In this Note we prove his conjecture in types \(B_n, C_n\) and \(D_n\) for adjoint case.

This paper is a contribution to the study of the subgroup structure of exceptional algebraic groups over algebraically closed fields of arbitrary characteristic. Following Serre, a closed subgroup of a semisimple algebraic group $G$ is called irreducible if it lies in no proper parabolic subgroup of $G$. In this paper we complete the classification of irreducible connected subgroups of exceptional algebraic groups, providing an explicit set of representatives for the conjugacy classes of such subgroups. Many consequences of this classification are also given. These include results concerning the representations of such subgroups on various $G$-modules: for example, the conjugacy classes of irreducible connected subgroups are determined by their composition factors on the adjoint module of $G$, with one exception. A result of Liebeck and Testerman shows that each irreducible connected subgroup $X$ of $G$ has only finitely many overgroups and hence the overgroups of $X$ form a lattice. We provide tables that give representatives of each conjugacy class of overgroups within this lattice structure. We use this to prove results concerning the subgroup structure of $G$: for example, when the characteristic is 2, there exists a maximal connected subgroup of $G$ containing a conjugate of every irreducible subgroup $A_1$ of $G$., Comment: 196 pages. To appear in Mem. Amer. Math. Soc

Let $\mathbb{k}$ be a field of characteristic $p$. We introduce a formalism of mixed sheaves with coefficients in $\mathbb{k}$ and showcase its use in representation theory. More precisely, we construct for all quasi-projective schemes $X$ over an algebraic closure of $\mathbb{F}_p$ a $\mathbb{k}$-linear triangulated category of motives on $X$. Using work of Ayoub (2007), Cisinski-Deglise (2012) and Geisser-Levine (2000), we show that this system of categories has a six functors formalism and computes higher Chow groups. Indeed, it behaves similarly to other categories of sheaves that one is used to. We attempt to make its construction also accessible to non-experts. We then consider the subcategory of stratified mixed Tate motives defined for affinely stratified varieties $X$, discuss perverse and parity motives and prove formality results. As an example, we combine these results and Soergel (2000) to construct a geometric and graded version of Soergel's modular category $\mathscr O(G)$, consisting of rational representations of a split semisimple group $G/\mathbb{k}$, and thereby equip it with a full six functor formalism (see Riche-Soergel-Williamson (2014) and Achar-Riche (2016) for other approaches). The main idea of using motives in geometric representation theory in this way as well as many results about stratified mixed Tate motives are directly borrowed from Soergel and Wendt, who tell the story in characteristic zero., Comment: New version, published in Selecta Mathematica New Series. (2019) 25: 30, see https://doi.org/10.1007/s00029-019-0475-x

Let $G$ be the group of $\mathbb R$--points of a semisimple algebraic group $\mathcal G$ defined over $\mathbb Q$. Assume that $G$ is connected and noncompact. We study Fourier coefficients of Poincar\' e series attached to matrix coefficients of integrable discrete series. We use these results to construct explicit automorphic cuspidal realizations, which have appropriate Fourier coefficients $\neq 0$, of integrable discrete series in families of congruence subgroups. In the case of $G=Sp_{2n}(\mathbb R)$, we relate our work to that of Li [15]. For $\mathcal G$ quasi--split over $\mathbb Q$, we relate our work to the result about Poincar\' e series due to Khare, Larsen, and Savin [16].