Spherical functions on orthogonal groups

Chapter 5 - Spherical functions of type X on a Riemannian symmetric space

Spherical functions on a real semisimple Lie group. A method of reduction to the complex case

Let G be a reductive algebraic group over Q and Γ⊂G(Q) an arithmetic subgroup. Let K∞⊂G(R) be a maximal compact subgroup. We study the asymptotic behavior of the counting functions of the cuspidal and residual spectrum, respectively, of the regular representation of G(R) in L2(Γ﹨G(R)) of a fixed K∞-type σ. A conjecture, which is due to Sarnak, states that the counting function of the cuspidal spectrum of type σ satisfies Weyl's law and the residual spectrum is of lower order growth. Using the Arthur trace formula we reduce the conjecture to a problem about L-functions occurring in the constant terms of Eisenstein series. If G satisfies property (L), introduced by Finis and Lapid, we establish the conjecture. This includes classical groups over a number field.

Let G G denote a connected reductive group over a nonarchimedean local field F F . Let K K denote a special maximal parahoric subgroup of G ( F ) G(F) . We establish a Satake isomorphism for the Hecke algebra H K \mathcal {H}_K of K K -bi-invariant compactly supported functions on G ( F ) G(F) . The key ingredient is a Cartan decomposition describing the double coset space K ∖ G ( F ) / K K\backslash G(F)/K . As an application we define a transfer homomorphism t : H K ∗ ( G ∗ ) → H K ( G ) t: \mathcal {H}_{K^*}(G^*) \rightarrow \mathcal {H}_K(G) where G ∗ G^* is the quasi-split inner form of G G . We also describe how our results relate to the treatment of Cartier [Car], where K K is replaced by a special maximal compact open subgroup K ~ ⊂ G ( F ) \widetilde {K} \subset G(F) and where a Satake isomorphism is established for the Hecke algebra H K ~ \mathcal {H}_{\widetilde {K}} .

Consider a reductive linear algebraic group G acting linearly on a polynomial ring S over an infinite field; key examples are the general linear group, the symplectic group, the orthogonal group, and the special linear group, with the classical representations as in Weyl’s book: For the general linear group, consider a direct sum of copies of the standard representation and copies of the dual; in the other cases, take copies of the standard representation. The invariant rings in the respective cases are determinantal rings, rings defined by Pfaffians of alternating matrices, symmetric determinantal rings and the Plücker coordinate rings of Grassmannians; these are the classical invariant rings of the title, with $S^G\subseteq S$ being the natural embedding. Over a field of characteristic zero, a reductive group is linearly reductive, and it follows that the invariant ring $S^G$ is a pure subring of S, equivalently, $S^G$ is a direct summand of S as an $S^G$ -module. Over fields of positive characteristic, reductive groups are typically no longer linearly reductive. We determine, in the positive characteristic case, precisely when the inclusion $S^G\subseteq S$ is pure. It turns out that if $S^G\subseteq S$ is pure, then either the invariant ring $S^G$ is regular or the group G is linearly reductive.

Many moduli spaces in complex algebraix geometry can be expressed as quotients, in the sense of Mumford’s geometric invariant theory [18], of nonsingular complex projective varieties X by actions of complex reductive groups G. Any such quotient can also be identified with a symplectic quotient (or Marsden-Weinstein reduction) of the variety X by a maximal compact subgroup K of the reductive group G [14], [18], [19]. This symplectic quotient is µ-1(0)/K, where µ : X → k* is a moment map for the action of K on X equipped with a suitable symplectic form.

This paper establishes a functorial algebraic isomorphism between the moduli space BCps(Σ,G) of polystable principal G-bundles with prescribed monodromy on a punctured Riemann surface Σ of genus g≥2, for a complex reductive Lie group G, and the character variety MCK(Σ*,G) of representations of its fundamental group with relatively compact image. The dimension formula dimBCps(Σ,G)=2(g−1)dimC(G)+∑i=1kdimR(Ci), where C1,…,Ck are conjugacy classes in a maximal compact subgroup K⊂G, is derived for complex reductive Lie groups, and singularities are characterized as polystable bundles with non-trivial automorphism groups. As applications of the above geometric results to control theory, it is proved that topologically distinct polystable robotic navigation strategies around obstacles are classified by this character variety. The geometry of singular points in families of polystable control strategies is further investigated, revealing enhanced stability properties characterized by reduced tangent space dimensions arising from non-trivial automorphism groups.

K-energy on polarized compactifications of Lie groups

The second degree cohomology of finite orthogonal groups, II

Let D = G / V D = G/V be a non-classical period domain, where G G is a semi-simple real Lie group with a compact Cartan subgroup T T and maximal compact subgroup K K , and V V the centralizer of a torus in G G . If the symmetric space is Hermitian, assume also that G / V G/V does not fiber holomorphically over G / K G/K . Let Γ \Gamma be a co-compact, torsion-free lattice in G G . We prove that the complex manifold Γ ∖ G / V \Gamma \backslash G/V does not admit any Kähler metric.

Let G be a connected real reductive group with maximal compact subgroup K of equal rank, and let ℳ be the category of Harish-Chandra modules for G. We relate three differently defined pairings between two finite length modules X and Y in ℳ: the Euler-Poincaré pairing, the natural pairing between the Dirac indices of X and Y, and the elliptic pairing of [2]. (The Dirac index I Dir (X) is a virtual finite-dimensional representation of K ˜, the spin double cover of K.) We construct index functions f X for any finite length Harish-Chandra module X. Each of these functions is very cuspidal in the sense of Labesse, and its orbital integral on elliptic elements coincides with the character of X. From this we deduce that the Dirac index pairing coincide with the elliptic pairing. Analogy with the case of Hecke algebras studied in [8] and [7] and a formal (but not rigorous) computation led us to conjecture that the first two pairings coincide. We show that they are both computed as the indices of Fredholm pairs (defined here in an algebraic sense) of operators acting on the same spaces. Recently, Huang and Sun have established the equality between the Euler-Poincaré and the elliptic pairing, thereby proving directly the analogue of a result of Schneider and Stuhler for p-adic groups [25].

A linear étale representation of a complex algebraic group G is given by a complex algebraic G-module V such that G has a Zariski-open orbit in V and $\dim G=\dim V$ . A current line of research investigates which reductive algebraic groups admit such étale representations, with a focus on understanding common features of étale representations. One source of new examples arises from the classification theory of nilpotent orbits in semisimple Lie algebras. We survey what is known about reductive algebraic groups with étale representations and then discuss two classical constructions for nilpotent orbit classifications due to Vinberg and to Bala and Carter. We determine which reductive groups and étale representations arise in these constructions and we work out in detail the relation between these two constructions.

Theory of Spherical Functions

Some remarks on the theory of spherical functions on symmetric Riemannian manifolds

LetSp(n, R) be the sympletic group, and letKn* be its maximal compact subgroup. ThenG=Sp(n,R)/Kn* can be realized as the Siegel domain of type one. The square-integrable representation ofG gives the admissible wavelets AW and wavelet transform. The characterization of admissibility condition in terms of the Fourier transform is given. The Bergman kernel follows from the viewpoint of coherent state. With the Laguerre polynomials, Hermite polynomials and Jacobi polynomials, two kinds of orthogonal bases for AW are given, and they then give orthogonal decompositions ofL2-space on the Siegel domain of type one ℒ(ℋn, |y| *dxdy).