Euler-Poincaré pairing, Dirac index and elliptic pairing for Harish-Chandra modules

The direct sum behaviour of its projective modules is a fundamental property of any ring. Hereditary Noetherian prime rings are perhaps the only noncommutative Noetherian rings for which this direct sum behaviour (for both finitely and infinitely generated projective modules) is well-understood, yet highly nontrivial . This book surveys material previously available only in the research literature. It provides a re-worked and simplified account, with improved clarity, fresh insights and many original results about finite length modules, injective modules and projective modules. It culminates in the authors' surprisingly complete structure theorem for projective modules which involves two independent additive invariants: genus and Steinitz class. Several applications demonstrate its utility. The theory, extending the well-known module theory of commutative Dedekind domains and of hereditary orders, develops via a detailed study of simple modules. This relies upon the substantial account of idealiser subrings which forms the first part of the book and provides a useful general construction tool for interesting examples. The book assumes some knowledge of noncommutative Noetherian rings, including Goldie's theorem. Beyond that, it is largely self-contained, thanks to the appendix which provides succinct accounts of Artinian serial rings and, for arbitrary rings, results about lifting direct sum decompositions from finite length images of projective modules. The appendix also describes some open problems.

An interpretation of the Casselman-Wallach Theorem is that the K K -finite functor is an isomorphism of categories from the category of finitely generated, admissible smooth Fréchet modules of moderate growth to the category of Harish-Chandra modules for a real reductive group, G G (here K K is a maximal compact subgroup of G G ). In this paper we study the dependence of the inverse functor to the K K -finite functor on parameters. Our main result implies that holomorphic dependence implies holomorphic dependence. The work uses results from the excellent thesis of van der Noort. Also a remarkable family of universal Harish-Chandra modules, developed in this paper, plays a key role.

We prove that the restriction to (2, R) of the underlying HarishChandra module of a Discrete Series representation of a noncompact simple Lie group has no sl(2)-irreducible subrepresentations unless it is holomorphic. We also give a characterization of the holomorphic Discrete Series. Let G be a connected real simple Lie group. Fix a maximal compact subgroup K of G. Let g (t) denote the complex Lie algebra of G (respectively, K). Hence we have the Cartan decomposition g = t + p. Thus, p is Kinvariant under the restriction of the adjoint representation of G to K and p . Let T C K be a maximal torus. Henceforth we assume that T is a Cartan subgroup of G. Let t be the complex Lie algebra of T. Under these hypotheses Harish-Chandra in [HC2] has given a classification of the square-integrable representations of G in terms of the regular characters of T. Let (D(g, t) denote the root system of the pair (g, t) . A root of the pair (g, t) is called compact (noncompact) if its root space is contained in t (respectively, p). We now fix a noncompact root y . Then the root vectors of y span a copy by of sl(2, C) . Moreover by is invariant under the conjugation of g with respect to the Lie algebra of G. Let s,, be the real form of by associated to the conjugation of g with respect the Lie algebra of G restricted to by . Let Hy be the connected subgroup of G corresponding to sy I Let (7r, V) be the Harish-Chandra module underlying a square-integrable, irreducible representation of G. We then have Proposition 1. If the restriction of (7r, V) to by contains an irreducible bysubmodule, then G/K is a Hermitian symmetric space and (7r, V) corresponds to a holomorphic square-integrable representation. A corollary to Proposition 1 is Corollary. If G is as above, G/K is not Hermitian, and H is a semisimple subgroup of G such that its symmetric space is Hermitian, then (7r, V) restricted to [ has no [-holomorphic irreducible subrepresentations. Received by the editors April 29, 1991 and, in revised form, July 11, 1991. 1991 Mathematics Subject Classification. Primary 22E47.

Let G be a p-adic connected reductive algebraic group and K a maximal compact subgroup of G. In [4], Casselman obtained the explicit formula of zonal spherical functions on G with respect to K on the assumption that K is special. It is known (Bruhat and Tits [3]) that the affine root system of algebraic group which has good but not special maximal compact subgroup is A1 C2, or Bn (n > 3), and all Bn-types can be realized by orthogonal groups. Here the assumption “good” is necessary for the Satake’s theory of spherical functions.

Let G be a real reductive algebraic group with maximal compact subgroup K, and let Fr be a rank r free group. We show that the space of closed orbits in Hom(Fr ,G)/G admits a strong deformation retraction to the orbit space Hom(Fr ,K)/K. In particular, all such spaces have the same homotopy type. We compute the Poincaré polynomials of these spaces for some low rank groups G, such as Sp(4,ℝ) and U(2,2). We also compare these real moduli spaces to the real points of the corresponding complex moduli spaces, and describe the geometry of many examples.

Let F be a non-Archimedean local field and let p be the residual characteristic of F. Let G=GL2(F) and let P be a Borel subgroup of G. In this paper we study the restriction of irreducible smooth representations of G on $\overline {\mathbf {F}}_{p}$-vector spaces to P. We show that in a certain sense P controls the representation theory of G. We then extend our results to smooth $\mathcal {O}[G]$-modules of finite length and unitary K-Banach space representations of G, where $\mathcal {O}$ is the ring of integers of a complete discretely valued field K with residue field $\overline {\mathbf {F}}_{p}$.

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.

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).

The first obstacle in building a Geometric Quantization theory for nilpotent orbits of a real semisimple Lie group has been the lack of an invariant polarization. In order to generalize the Fock space construction of the quantum mechanical oscillator, a polarization of the symplectic orbit invariant under the maximal compact subgroup is required.

We give a general bosonic construction of oscillator-like unitary irreducible representations (UIR) of non-compact groups whose coset spaces with respect to their maximal compact subgroups are Hermitian symmetric. With the exception of E7(7), they include all the non-compact invariance groups of extended supergravity theories in four dimensions. These representations have the remarkable property that each UIR is uniquely determined by an irreducible representation of the maximal compact subgroup. We study the connection between our construction, the Hermitian symmetric spaces and the Tits-Koecher construction of the Lie algebras of corresponding groups. We then give the bosonic construction of the Lie algebra ofE 7(7) in SU(8), SO(8) and U(7) bases and study its properties. Application of our method toE 7(7) leads to reducible unitary representations.

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.

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

Self dual symmetric $R$-spaces have special curves, called circles, introduced by Burstall, Donaldson, Pedit and Pinkall in 2011, whose definition does not involve the choice of any Riemannian metric. We characterize the elements of the big transformation group $G$ of a self dual symmetric $R$-space $M$ as those diffeomorphisms of $M$ sending circles in circles. Besides, although these curves belong to the realm of the invariants by $G$, we manage to describe them in Riemannian geometric terms: Given a circle $c$ in $M$, there is a maximal compact subgroup $K$ of $G$ such that $c$, except for a projective reparametrization, is a diametrical geodesic in $M$ (or equivalently, a diagonal geodesic in a maximal totally geodesic flat torus of $M$), provided that $M$ carries the canonical symmetric $K$-invariant metric. We include examples for the complex quadric and the split standard or isotropic Grassmannians.

On the types for supercuspidal representations of inner forms of GLN

We prove that the asymptotic Assouad-Nagata dimension of a connected Lie group $G$ equipped with a left-invariant Riemannian metric coincides with its topological dimension of $G/C$ where $C$ is a maximal compact subgroup. To prove it we will compute the Assouad-Nagata dimension of connected solvable Lie groups and semisimple Lie groups. As a consequence we show that the asymptotic Assouad-Nagata dimension of a polycyclic group equipped with a word metric is equal to its Hirsch length and that some wreath-type finitely generated groups can not be quasi-isometric to any cocompact lattice on a connected Lie group.