Branching of Representations to Symmetric Subgroups
Michael G. Eastwood, Joseph A. Wolf
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28 pages, with a number of LiE programs for branching of representations to subgroups defined by automorphism, while keeping track of the action on both the center and the semisimple part of the subgroup
Let $\gg$ be the Lie algebra of a compact Lie group and let $θ$ be any automorphism of $\gg$. Let $\gk$ denote the fixed point subalgebra $\gg^θ$. In this paper we present LiE programs that, for any finite dimensional complex representation $π$ of $\gg$, give the explicit branching $π|_\gk$ of $π$ on $\gk$. Cases of special interest include the cases where $θ$ has order 2 (corresponding to compact riemannian symmetric spaces $G/K$), where $θ$ has order 3 (corresponding to compact nearly--kaehler homogeneous spaces $G/K$), where $θ$ has order 5 (which include the fascinating 5--symmetric space $E_8/A_4A_4$), and the cases where $\gk$ is the centralizer of a toral subalgebra of $\gg$.
We construct polyhedral families of valuations on the branching algebra of a morphism of reductive groups. This establishes a connection between the combinatorial rules for studying a branching problem and the tropical geometry of the branching algebra. In the special case when the branching problem comes from the inclusion of a Levi subgroup or a diagonal subgroup, we use the dual canonical basis of Lusztig and Kashiwara to build toric deformations of the branching algebra.
We study branching problem of the metaplectic representation of $Sp(2, \mathbb R)$ under its principle subgroup $SL(2, \mathbb R)$. We find the complete decomposition.
Let g be a complex semi-simple Lie algebra and g be a semisimple subalgebra of g. Consider the branching problem of decomposing the simple g-representations V as a sum of simple grepresentations V. When g = g x g, it is the tensor product decomposition. The multiplicity space Mult(V, V) satisfies V = $\oplus$ V Mult(V, V) $\otimes$ V, where the sum runs over the isomorphism classes of simple g-representations. In the case when g is spherical of minimal rank, we describe Mult(V, V) as the intersection of kernels of powers of root operators in some weight space of the dual space V * of V. When g…
We give a branching law for subgroups fixed by an involution. As an application we give a generalization of the Cartan-Helgason theorem and a noncompact analogue of the Borel-Weil theorem.
We introduce the notion of ideal triangle in the Bruhat-Tits building associated to a split group -- it is analogous to the usual notion of triangle, but one vertex is "at infinity" in a certain direction. We prove that the algebraic variety of based ideal triangles with prescribed side-lengths is naturally isomorphic to a suitable variety of genuine triangles. From theorems pertaining to genuine triangles, we deduce saturation theorems related to branching to Levi subgroups and to the constant term homomorphisms.
Let $G$ be a connected reductive subgroup of a complex connected reductive group $\hat{G}$. We are interested in the branching problem. Fix maximal tori and Borel subgroups of $G$ and $\hat G$. Consider the cone $lr(G,\hat G)$ generated by the pairs $(ν,\hat nu)$ of dominant characters such that $V_ν^*$ is a submodule of $V_{\hat nu}$. It is known that $lr(G,\hat G)$ is a closed convex polyhedral cone. In this work, we show that every regular face of $lr(G,\hat G)$ gives rise to a {\it reduction rule} for multiplicities. More precisely, we prove that for $(ν,\hat nu)$ on such a face, the multi…
Given $Γ$ a finite subgroup of $\mathbf{SL}_3\mathbb{C}$, we determine how an arbitrary finite dimensional irreducible representation of $\mathbf{SL}_3\mathbb{C}$ decomposes under the action of $Γ$. To the subgroup $Γ$ we attach a generalized Cartan matrix $C_Γ$. Then, inspired by B. Kostant, we decompose the Coxeter element of the Kac-Moody algebra attached to $C_Γ$ as a product of reflections of a special form, thereby suggesting an algebraic form for the McKay correspondence in dimension 3.
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 …
The purpose of this paper is to prove Duflo's conjecture for $(G,π, AN)$ where $G$ is a simple Lie group of Hermitian type and $π$ is a discrete series of $G$ and $AN$ is the maximal exponential solvable subgroup for an Iwasawa decomposition $G=KAN$. This is essentially reduced from the following general theorem we prove in this paper: let $G$ be a connected semisimple Lie group . Then a strongly elliptic $G$-coadjoint orbit $\mathcal{O}$ is holomorphic if and only if $\text{p}(\mathcal{O})$ is an open $AN$-coadjoint orbit, where $\text{p} : \mathfrak{g}^* \longrightarrow (\mathfrak{a}\oplus\m…
We study the restriction to Sylow subgroups of irreducible characters of symmetric groups. In particular, we focus our attention on constituents of degree greater than 1. Our main result is a wide generalization of Theorem 3.1 of Giannelli and Navarro, "Restricting irreducible characters to Sylow p-subgroups".
The unitary principal series representations of $G=GL(n,\mathbb{C})$ induced from a character of the maximal parabolic subgroup $P=(GL(1,\mathbb{C})\times GL(n-1,\mathbb{C}))\ltimes\mathbb{C}^{n-1}$ attain the minimal Gelfand--Kirillov dimension among all infinite-dimensional unitary representations of $G$. We find the explicit branching laws for the restriction of these representations to symmetric subgroups of $G$.
In this paper, we obtain explicit branching laws for all irreducible unitary representations of $\Spin(N,1)$ restricted to a parabolic subgroup $P$. The restriction turns out to be a finite direct sum of irreducible unitary representations of $P$. We also verify Duflo's conjecture for the branching law of tempered representations of $\Spin(N,1)$ with respect to a parabolic subgroup $P$. That is to show: in the framework of the orbit method, the branching law of a tempered representation is determined by the behavior of the moment map from the corresponding coadjoint orbit. A few key tools used…
Given a complex simply connected simple algebraic group $G$ of exceptional type and a maximal parabolic subgroup $P \subset G$, we classify all triples $(G,P,H)$ such that $H \subset G$ is a maximal reductive subgroup acting spherically on $G/P$. In addition we derive branching rules for $\text{res}^G_H (V^*_{kω_i})$, $k \in \N$, where $ω_i$ is the fundamental weight associated to $P$. This is the first of two parts of a project to classify all such triples and corresponding branching rules for all simply connected simple algebraic groups.
Let $G=\operatorname{O}(1,n+1)$ with maximal compact subgroup $K$ and let $Π$ be a unitary irreducible representation of $G$ with non-trivial $(\mathfrak{g},K)$-cohomology. Then $Π$ occurs inside a principal series representation of $G$, induced from the $\operatorname{O}(n)$-representation $\bigwedge\nolimits^p(\mathbb{C}^n)$ and characters of a minimal parabolic subgroup of $G$ at the limit of the complementary series. Considering the subgroup $G'=\operatorname{O}(1,n)$ of $G$ with maximal compact subgroup $K'$, we prove branching laws and explicit Plancherel formulas for the restrictions to…
We decompose the restriction of ramified principal series representations of the $p$-adic group $\mathrm{GL}(3,\mathrm{k})$ to its maximal compact subgroup $K=\mathrm{GL}(3,\mathscr{R})$. Its decomposition is dependent on the degree of ramification of the inducing characters and can be characterized in terms of filtrations of the Iwahori subgroup in $K$. We establish several irreducibility results and illustrate the decomposition with some examples.
Let $G$ be a connected reductive algebraic group over an algebraically closed field ${\bf k}$ of characteristic not equal to 2, let $\B$ be the variety of all Borel subgroups of $G$, and let $K$ be a symmetric subgroup of $G$. Fixing a closed $K$-orbit in $\B$, we associate to every $K$-orbit on $\B$ some subsets of the Weyl group of $G$, and we study them as invariants of the $K$-orbits. When ${\bf k} = {\mathbb C}$, these invariants are used to determine when an orbit of a real form of $G$ and an orbit of a Borel subgroup of $G$ have non-empty intersection in $\B$. We also characterize the i…
We give a description of the irreducible constituents of the restriction to Sylow 2-subgroups of irreducible characters of symmetric groups labelled by hook partitions.
In the framework of McKay correspondence we determine, for every finite subgroup $Γ$ of $\mathbf{SL}_4\mathbb{C}$, how the finite dimensional irreducible representations of $\mathbf{SL}_4\mathbb{C}$ decompose under the action of $Γ$. Let $\go{h}$ be a Cartan subalgebra of $\go{sl}_4\mathbb{C}$ and let $\varpi_1,\,\varpi_2,\,\varpi_3$ be the corresponding fundamental weights. For $(p,q,r)\in \mathbb{N}^3$, the restriction $π_{p,q,r}|_Γ$ of the irreducible representation $π_{p,q,r}$ of highest weight $p\varpi_1+q\varpi_2+r\varpi_3$ of $\mathbf{SL}_4\mathbb{C}$ decomposes as ${π_{p,q,r}}|_Γ=\bigo…
We approach the problem of obtaining branching rules from the point of view of dual reductive pairs. Specifically, we obtain a stable branching rule for each of 10 classical families of symmetric pairs. In each case, the branching multiplicities are expressed in terms of Littlewood-Richardson coefficients. Some of the formulas are classical and include, for example, Littlewood's restriction rule as a special case.
Let $G$ be a symplectic or special orthogonal group, let $H$ be a connected reductive subgroup of $G$, and let $X$ be a flag variety of $G$. We classify all triples $(G,H,X)$ such that the natural action of $H$ on $X$ is spherical. For each of these triples, we determine the restrictions to $H$ of all irreducible representations of $G$ realized in spaces of sections of homogeneous line bundles on $X$.
For a connected, noncompact matrix simple Lie group $G$ so that a maximal compact subgroup $K$ has three dimensional simple ideal, in this note we analyze the admissibility of the restriction of irreducible square integrable representations for the ambient group when they are restricted to certain subgroups that contains the three dimensional ideal. In this setting we provide a formula for the multiplicity of the irreducible factors. Also, for general $G$ such that $G/K$ is an Hermitian $G$-manifold we give a necessary and sufficient condition so that a square integrable representations of the…
Let $Γ$ be a finite subgroup of SU(2) and let $\widetilde Γ = \{γ_i\mid i\in J\}$ be the unitary dual of $Γ$. The unitary dual of SU(2) may be written $\{π_n\mid n\in \Bbb Z_+\}$ where $dim π_n = n+1$. For $n\in \Bbb Z_+$ and $j\in J$ let $m_{n,j}$ be the multiplicity of $γ_j$ in $π_n|Γ$. Then we collect this branching data in the formal power series, $m(t)_j = \sum_{n=0}^{\infty}m_{n,j} t^n$. One shows that there exists a polynomial $z(t)_j$ and known positive integers $a,b$ (independent of $j$) such that $m(t)_j = {z(t)_j \over (1-t^a)(1-t^b)}$. The problem is the determination of the polyno…
We review a class of modules for the wreath product S(m) wr S(n) of two symmetric groups which are analogous to the Specht modules of the symmetric group, and prove a pair of branching rules for this family of modules. These branching rules describe the behaviour of these wreath product Specht modules under restriction to the wreath products S(m-1) wr S(n) and S(m) wr S(n-1). In particular, we see that these restrictions of wreath product Specht modules have Specht module filtrations, and we obtain combinatorial interpretations of the multiplicities in these filtrations.
We show that the Sylow $p$-subgroups of a symmetric group, respectively an alternating group, are characterized as the $p$-subgroups containing all elementary abelian $p$-subgroups up to conjugacy of the symmetric group, respectively the alternating group. We apply the characterization result for symmetric groups to compute the vertices of the hook Specht modules associated to the partition $(kp-p,1^p)$ under the assumption that $k \equiv 1$ mod $p$ and $k \not\equiv 1$ mod $p^2$.
We give a modular branching rule for certain wreath products as a generalization of Kleshchev's modular branching rule for the symmetric groups. Our result contains a modular branching rule for the complex reflection groups $G(m,1,n)$ (which are often called the generalized symmetric groups) in splitting fields for $\mathbb{Z}/m\mathbb{Z}$. Especially for $m=2$ (which is the case of the Weyl groups of type $B$), we can give a modular branching rule in any field. Our proof is elementary in that it is essentially a combination of Frobenius reciprocity, Mackey theorem, Clifford's theory and Klesh…
A reduction formula for the branching coefficients of tensor products of representations and more generally restrictions of representations of a semisimple group to a semisimple subgroup is proved in work by Knutson-Tao and Derksen-Weyman. This formula holds when the highest weights of the representations belong to a codimension 1 face of the Horn cone, which by work by Ressayre corresponds to a Littlewood-Richardson coefficient equal to 1. We prove a similar reduction formula when this coefficient is equal to 2, and show some properties of the class of the branch divisor corresponding to a ge…
We study branching laws for a classical group $G$ and a symmetric subgroup $H$. Our approach is through the {\it branching algebra}, the algebra of covariants for $H$ in the regular functions on the natural torus bundle over the flag manifold for $G$. We give concrete descriptions of (natural subalgebras of) the branching algebra using classical invariant theory. In this context, it turns out that the ten classes of classical symmetric pairs $(G,H)$ are associated in pairs, $(G,H)$ and $(H',G')$, and that the (partial) branching algebra for $(G,H)$ also describes a branching law from $H'$ to $…
In this article we investigate the positivity of Sylow branching coefficients for symmetric groups when $p = 3$. In particular, we complete the discussion begun by Giannelli and the author in arXiv:1712.02642 (J. Algebra) and developed in arXiv:1909.09446 (J. London Math. Soc.) concerning the case of odd primes.
Symmetric $k$-varieties are a natural generalization of symmetric spaces to general fields $k$. We study the action of minimal parabolic $k$-subgroups on symmetric $k$-varieties and define a map that embeds these orbits within the orbits corresponding to algebraically closed fields. We develop a condition for the surjectivity of this map in the case of $k$-split groups that depends only on the dimension of a maximal $k$-split torus contained within the fixed point group of the involution defining the symmetric $k$-variety.
Let $\widetilde{G}$ be the $n$-fold covering group of the special linear group of degree two, over a non-Archimedean local field. We determine the decomposition into irreducibles of the restriction of the principal series representations of $\widetilde{G}$ to a maximal compact subgroup of $\widetilde{G}$.
In this paper we consider the unitary symmetric spaces of the form X=U(p,q)/U(1)U(p,q-1) and their discrete series representations. Inspired by the work of A.Venkatesh and Y.Sekellarides on L-groups of p-adic spherical spaces we formulate and prove natural relative branching laws for the restriction to smaller subgroups of the same type and corresponding unitary spaces.We think of this as steps to formulation and proving Gan Gross Prasad conjectures for unitary spaces. Using period integral and some results of T.Kobayashi we prove an analogue of thesis conjectures.
In this paper we consider the restriction of a unitary irreducible representation of type $A_{\mathfrak q}(λ)$ of $GL(4,{\mathbb R})$ to reductive subgroups $H$ which are the fixpoint sets of an involution. We obtain a formula for the restriction to the symplectic group and to $GL(2,{\mathbb C})$, and as an application we construct in the last section some representations in the cuspidal spectrum of the symplectic and the complex general linear group. In addition to working directly with the cohmologically induced module to obtain the branching law, we also introduce the useful concept of pseu…
Let $p\ge 5$ be a prime and let $n$ be a natural number. In this article we describe the irreducible constituents of the induced characters $ϕ\big\uparrow^{\mathfrak{S}_n}$ for arbitrary linear characters $ϕ$ of a Sylow $p$-subgroup of the symmetric group $\mathfrak{S}_n$, generalising earlier results of the authors. By doing so, we introduce Sylow branching coefficients for symmetric groups.
The main result of this paper is a new parameterization of the orbits of a symmetric subgroup on a partial flag variety. The parameterization is in terms of certain Spaltenstein varieties, on one hand, and certain nilpotent orbits, on the other. One of the motivations is related to enumerating special unipotent representations of real reductive groups. Another motivation is understanding (a portion of) the closure order on the set of nilpotent coadjoint orbits.
Let $G$ be a split reductive group over a $p$-adic field $F$. Let $B$ be a Borel subgroup and $U$ the maximal unipotent subgroup of $B$. Let $ψ$ be a Whittaker character of $U$. Let $I$ be an Iwahori subgroup of $G$. We describe the Iwahori-Hecke algebra action on the Gelfand-Graev representation $(\mathrm{ind}_{U}^{G}ψ)^I$ by an explicit projective module. As a consequence, for $G=GL(n,F)$, we define and describe Bernstein-Zelevinsky derivatives of representations generated by $I$-fixed vectors in terms of the corresponding Iwahori-Hecke algebra modules. Furthermore, using Lusztig's reduction…
The irreducible decomposition of a unitary representation often contains continuous spectrum when restricted to a non-compact subgroup. The author singles out a nice class of branching problems where each irreducible summand occurs discretely with finite multiplicity (admissible restrictions). Basic theory and new perspectives of admissible restrictions are presented from both analytic and algebraic view points. We also discuss some applications of admissible restrictions to modular varieties and $L^p$-harmonic analysis.
Let F be a local field of character zero. Let E be a quadratic field extension of F. We show that any P-invariant linear functional on a GL(n,E)-distinguished irreducible smooth admissible representation of GL(2n,F) is also GL(n,E)-invariant where P is a mirabolic subgroup of GL(n,E).
In this paper branching rules for the fundamental representations of the symplectic groups in positive characteristic are found. The submodule structure of the restrictions of the fundamental modules for the group $Sp_{2n}(K)$ to the naturally embedded subgroup $Sp_{2n-2}(K)$ is determined. As a corollary, inductive systems of fundamental representations for $Sp_{\infty}(K)$ are classified. The submodule structure of the fundamental Weyl modules is refined.
On restriction to the maximal compact subgroup $\mathrm{GL}(3,\mathscr{R})$, an unramified principal series representation of the $p$-adic group $\mathrm{GL}(3,F)$ decomposes into a direct sum of finite-dimensional irreducibles each appearing with finite multiplicity. We describe a coarser decomposition into components which, although reducible in general, capture the equivalences between the irreducible constituents.
In this paper we study the Plancherel formula for a new class of homogeneous spaces for real reductive Lie groups; these spaces are fibered over non-Riemannian symmetric spaces, and they exhibit a phenomenon of uniform infinite multiplicities. They also provide examples of non-tempered representations of the group appearing in the Plancherel formula. Several classes of examples are given.
We derive an asymptotic expansion for the subgroup of arbitrary Fuchsian groups and some other classes of large groups. Moreover, the main conjecture for Random Walks on symmetric groups is established in full generality. Both problems require subtle estimates of character values and root number multiplicities for symmetric groups, which are also of independent interest.
Let $G$ be a complex reductive group and $H=G^θ$ be its fixed point subgroup under a Galois involution $θ$. We show that any $H$-distinguished representation $π$ (i.e $\mathrm{dim}_{\mathbb{C}}\left(π^{*}\right)^{H}\neq0$) satisfies: 1) $π^θ\simeq\tildeπ$, where $\tildeπ$ is the contragredient representation and $π^θ$ is the twist of $π$ under $θ$. 2) $\mathrm{dim}_{\mathbb{C}}\left(π^{*}\right)^{H}\leq\left|B\backslash G/H\right|$, where $B$ is a Borel subgroup of $G$. By proving Statement 1), we give a partial answer to a conjecture by Lapid.
In this paper we study branching laws for certain unitary representations. This is done on the smooth vectors via a version of the {\it period integrals}, studied in number theory, and also closely connected to the {\it symmetry-breaking operators}, introduced by T.~Kobayashi. We exhibit non-vanishing symmetry breaking operators for the restriction of a representation $Π$ in the discrete spectrum for real hyperboloids to representations of smaller orthogonal groups. In the last part we discuss some conjectures for the restriction of representations in Arthur packets containing the representati…
We study natural quantizations of branching coefficients corresponding to the restrictions of the classical Lie groups to their Levi subgroups. We show that they admit a stable limit which can be regarded as a $q$-analogue of a tensor product multiplicity. According to a conjecture by Shimozono, the stable one-dimensional sum for nonexceptional affine crystals are expected to occur as special cases of these $q$-analogues.
If $G$ is a reductive Lie group of Harish-Chandra class, $H$ is a symmetric subgroup, and $π$ is a discrete series representation of $G$, the authors give a condition on the pair $(G,H)$ which guarantees that the direct integral decomposition of $π|_H$ contains each irreducible representation of $H$ with finite multiplicity. In addition, if $G$ is a reductive Lie group of Harish-Chandra class, and $H\subset G$ is a closed, reductive subgroup of Harish-Chandra class, the authors show that the multiplicity function in the direct integral decomposition of $π|_H$ is constant along `continuous para…
We consider the spherical complementary series of rank one Lie groups $H_n=\SO_0(n, 1; \mathbb F)$ for $\mathbb F=\mathbb R, \mathbb C, \mathbb H$. We prove that there exist finitely many discrete components in its restriction under the subgroup $H_{n-1}=\SO_0(n-1, 1; \mathbb F)$. This is proved by imbedding the complementary series into analytic continuation of holomorphic discrete series of $G_n=SU(n, 1)$, $SU(n, 1)\times SU(n, 1)$ and $SU(2n, 2)$ and by the branching of holomorphic representations under the corresponding subgroup $G_{n-1}$.
We use binary trees to study the Bratteli diagram of Sylow 2-subgroups of symmetric groups. We show that it is simple, has a recursive structure, and self-similarities at all scales. We contrast its subgraph of one-dimensional representations with the Macdonald tree. We exploit the recursive structure to find the multiplicities of irreducible characters in the restriction to a Sylow 2-subgroup of odd-dimensional representations of the symmetric group $S_{2^k}$.
The restriction of a supercuspidal representation of SL_2(k), for k a local nonarchimedean field, to a maximal compact subgroup decomposes as a multiplicity-free direct sum of irreducible representations. We explicitly describe this decomposition in the case that the residual characteristic is odd, and determine how the spectrum of this decomposition varies as a function of the parameters describing the supercuspidal representation.
