Spherical Subgroups Research Articles

Branching Coefficients of Holomorphic Representations and Segal–Bargmann Transform

Let D = G/ K be a complex bounded symmetric domain of tube type in a Jordan algebra V C , and let D= H/ L= D ∩ V be its real form in a Jordan algebra V⊂ V C . The analytic continuation of the holomorphic discrete series on Dopf; forms a family of interesting representations of G. We consider the restriction on D and the branching rule under H of the scalar holomorphic representations. The unitary part of the restriction map gives then a generalization of the Segal–Bargmann transform. The group L is a spherical subgroup of K and we find a canonical basis of L-invariant polynomials in the components of the Schmid decomposition and we express them in terms of the Jack symmetric polynomials. We prove that the Segal–Bargmann transform of those L-invariant polynomials are, under the spherical transform on D, multi-variable Wilson-type polynomials and we give a simple alternative proof of their orthogonality relation. We find the expansion of the spherical functions on D, when extended to a holomorphic function in a neighborhood of 0∈ D , in terms of the L-spherical holomorphic polynomials on D , the coefficients being the Wilson polynomials.

Orbits and Invariants Associated with a Pair of Spherical Varieties: Some Examples

Let H and K be spherical subgroups of a reductive complex group G. In many cases, detailed knowledge of the double coset space H\G/K is of fundamental importance in group theory and representation theory. If H or K is parabolic, then H\G/K is finite, and we recall the classification of the double cosets in several important cases. If H=K is a symmetric subgroup of G, then the double coset space K\G/K (and the corresponding invariant theoretic quotient) are no longer finite, but several nice properties hold, including an analogue of the Chevalley restriction theorem. These properties were generalized by Helminck and Schwarz (Duke Math. J.106(2) (2001), pp. 237–279) to the case where H and K are fixed point groups of commuting involutions. We recall Helminck and Schwarz's main results. We also give examples to show the difficulty in extending these results if we allow H=K to be a reductive spherical (nonsymmetric) subgroup or if we have H symmetric and K spherical reductive.

On orbit closures of spherical subgroups in flag varieties

Let \cal F be the flag variety of a complex semi-simple group G, let H be an algebraic subgroup of G acting on \cal F with finitely many orbits, and let V be an H-orbit closure in \cal F . Expanding the cohomology class of V in the basis of Schubert classes defines a union V0 of Schubert varieties in \cal F with positive multiplicities. If G is simply-laced, we show that these multiplicities are equal to the same power of 2. For arbitrary G, we show that V0 is connected in codimension 1. If moreover all multiplicities are 1, we show that the singularities of V are rational and we construct a flat degeneration of V to V0 in \cal F . Thus, for any effective line bundle L on \cal F , the restriction map {H}^0({\cal F}, {\rm L}) \to {H}^0({\rm V, L} is surjective, and {H}^n({\rm V, L}) = 0 for all n \geq 1 .

Finiteness of Double Coset Spaces

This paper makes a contribution to the classification of reductive spherical subgroups of simple algebraic groups over algebraically closed fields (a subgroup is called spherical if it has finitely many orbits on the flag variety). The author produces a class of reductive subgroups, and shows that in positive characteristic any such is spherical. The class includes all centralizers of inner involutions if the characteristic is odd, for which the result was already known thanks to work of Springer; however, the sphericality of the corresponding subgroups in even characteristic was in many cases previously undecided. The methods used are character-theoretic in nature, and rely upon bounding inner products of permutation characters. 1991 Mathematics Subject Classification: primary 20G15; secondary 20C15.

On deformation method in invariant theory

In this paper we relate the deformation method in invariant theory to spherical subgroups. Let G be a reductive group, Z an affine G-variety and H⊂G a spherical subgroup. We show that whenever G/H is affine and its semigroup of weights is saturated, the algebra of H-invariant regular functions on Z has a G-invariant filtration such that the associated graded algebra is the algebra of regular functions of some explicit horospherical subgroup of G. The deformation method in its usual form, as developed by Luna et al., is a particular case of this construction. Our result also applies to the description of invariants of some reducible representations of reductive groups.

On Irreducible Representations of So2n+1 × So2m

AbstractIn this paper, we study the restriction of irreducible representations of the group SO2n+1 × SO2m to a spherical subgroup.

Restriction of Discrete Series of SU(2,1) to S( U(1) × U(1,1))

The group G = SU(2, 1) possesses nonempty holomorphic, antiholomorphic, and nonholomorphic discrete series. The restriction of these discrete series to the spherical subgroup G 1 = S( U(1) × U(1, 1)) is studied in this paper. We prove that direct integral decomposition of any restricted discrete series of G is multiplicity free. In ( J. Funct. Anal. 103 (1992), 352-371), J. Vargas claimed that there was no discrete part in the direct integral of any restricted nonholomorphic discrete series. Unfortunately, his proof was wrong. Our argument in Section 5 shows that there are infinitely many discrete series of G 1 occurring in the discrete part for any restricted nonholomorphic discrete series of G, and both the discrete and the continuous parts are not empty. The decomposition in Section 5 confirms a conjecture of B. Gross (B. H. Gross and D. Prasad, Canad. J. Math. 44, No. 5 (1992), 974-1002) for these groups. Our main interest is of course in the restriction of nonholomorphic discrete series, but for completeness, we consider all discrete series of G.

On Lie algebra decompositions related to spherical homogeneous spaces

LetG be a connected, reductive, linear algebraic group over an algebraically closed fieldk of characteristik zero. LetH 1 andH 2 be two spherical subgroups ofG. It is shown that for allg in a Zariski open subset ofG one has a Lie algebra decomposition g = h1 + Adg ⋅ h2, where a is the Lie algebra of a torus and dim a ≤ min (rankG/H 1,rankG/H 2). As an application one obtains an estimate of the transcendence degree of the fieldk(G/H 1 xG/H 2) G for the diagonal action ofG. Ifk = ℂ andG a is a real form ofG defined by an antiholomorphic involution σ :G→G then for a spherical subgroup H ⊂ G and for allg in a Hausdorff open subset ofG one has a decomposition g = ga + a Adg ⋅ h, where a is the Lie algebra of σ-invariant torus and dim a ≤ rankG/H.

A generic $H$-invariant in a multiplicity-free $G$-action

For a prehomogeneous action of a reductive group G on a vector space V, we construct a formal power series L that is shown to have a nonzero projection to every G-isotypic component of P(V). When (G, V) is multiplicity-free, these Fourier components of the function L provide all the H-invariants in P(V) for some spherical subgroup H of G. Three interesting examples are presented. I. MAIN RESULT Let G be a connected reductive complex algebraic group, G D CV . Suppose that G acts on a complex vector space V prehomogeneously, i.e., G has a Zariski open (hence dense) orbit in V. From [Se], G acts on V* prehomogeneously. Fix 0 $& q in the open G-orbit in V*. Consider L = en = >n>0 q/n! as a formal power series on V. Let P(V) = > P(V), afEG be the G-isotypic decomposition of P( V), the algebra of polynomial functions on V. Theorem 1. L has a nonzero projection to all P(V)q such that P(V), :$ 0. Let H be the stabilizer in G of 0. By definition, b is an H-invariant, as is the projection of L = en to P(V)q . Let L ZLa aEG be the G-isotypic decomposition of L, thus we have 0 :$ La e P(V)q n P(V)H, if P(V)a $O. Now suppose the action of G on V is multiplicity-free, namely, the natural action of G on P(V) is multiplicity-free as a representation of G. From [Se, VK], G acts on V (V*, resp.) prehomogeneously. Let q, H, L be defined as in above. In this case, since H C G is spherical (see [HU] for example), i.e., dimc ,H < 1 for any irreducible finite-dimensional representation of G, and Received by the editors October 23, 1990 and, in revised form, December 18, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 20G20, 15A72. Q) 1992 American Mathematical Society 0002-9939/92 $1.00 + $.25 per page

Complexity and rank of homogeneous spaces

We study an algebraic varieties with the action of a reductive group G. The relation is elucidated between the notions of complexity and rank of an arbitrary G-variety and the structure of stabilizers of general position of some actions of G itself and its Borel subgroup. The application of this theory to homogeneous spaces provides the explicit formulas for the rank and the complexity of quasiaffine G/H in terms of co-isotropy representation of H. The existence of Cartan subspace (and hence the freeness of algebra of invariants) for co-isotropy representation of a connected observable spherical subgroup H is proved.

A note on spherical function-induced subgroups

A note on spherical function-induced subgroups

Homogeneous domains on flag manifolds and spherical subgroups of semisimple Lie groups

Homogeneous domains on flag manifolds and spherical subgroups of semisimple Lie groups