Theorem Of Harish-Chandra Research Articles

Geometric approach to parabolic induction

In this paper we construct a “restriction” map from the cocenter of a reductive group G over a local non-archimedean field F to the cocenter of a Levi subgroup. We show that the dual map corresponds to parabolic induction and deduce that parabolic induction preserves stability. We also give a new (purely geometric) proof that the character of normalized parabolic induction does not depend on the parabolic subgroup. In the appendix, we use a similar argument to extend a theorem of Lusztig–Spaltenstein on induced unipotent classes to all infinite fields. We also prove a group version of a theorem of Harish-Chandra about the density of the span of regular semisimple orbital integrals.

Local integrability results in harmonic analysis on reductive groups in large positive characteristic

Let G be a connected reductive algebraic group over a non-Archimedean local field K, and let g be its Lie algebra. By a theorem of Harish-Chandra, if K has characteristic zero, the Fourier transforms of orbital integrals are represented on the set of regular elements in g(K) by locally constant functions, which, extended by zero to all of g(K), are locally integrable. In this paper, we prove that these functions are in fact specializations of constructible motivic exponential functions. Combining this with the Transfer Principle for integrability of [8], we obtain that Harish-Chandra's theorem holds also when K is a non-Archimedean local field of sufficiently large positive characteristic. Under the hypothesis that mock exponential map exists, this also implies local integrability of Harish-Chandra characters of admissible representations of G(K), where K is an equicharacteristic field of sufficiently large (depending on the root datum of G) characteristic.

QUANTUM GROUPS AT ROOTS OF UNITY AND MODULARITY

We develop the basic representation theory of all quantum groups at all roots of unity (that is, for q any root of unity, where q is defined as in [18]), including Harish–Chandra's theorem, which allows us to show that an appropriate quotient of a subcategory gives a semisimple ribbon category. This work generalizes previous work on the foundations of representation theory of quantum groups at roots of unity which applied only to quantizations of the simplest groups, or to certain fractional levels, or only to the projective form of the group. The second half of this paper applies the representation theory to give a sequence of results crucial to applications in topology. In particular, for each compact, simple, simply-connected Lie group we show that at each integer level the quotient category is in fact modular (thus leading to a Topological Quantum Field Theory), we determine when at fractional levels the corresponding category is modular, and we give a quantum version of the Racah formula for the decomposition of the tensor product.

Invariant hyperkaehler structures on the cotangent bundles of Hermitian symmetric spaces

Let G/K be an irreducible Hermitian symmetric space of compact type with standard homogeneous complex structure. Then the real symplectic manifold (T*(G/K),{omega}) has the natural complex structure J{sup -}. All G-invariant Kehler structures (J,{omega}) on G-invariant subdomains of T*(G/K) anticommuting with J{sup -} are constructed. Each hypercomplex structure of this kind, equipped with a suitable metric, defines a hyperkehler structure. As an application, a new proof of the theorem of Harish-Chandra and Moore for Hermitian symmetric spaces is obtained.

Une condition nécessaire et suffisante d'existence de la série discrète d'un groupe localement compact

Let G be a topological locally compact group. The aim of this Note is a contribution to the study of the existence problem for square integrable continuous and unitary representations for G. One of our main results (Theorem 6.2) will give a necessary and sufficient condition for the existence of the discrete series for G. Our approach is based on the notions of units and bounded elements in L 2( G) introduced by R. Godement in [6]. We perform a study of these notions. A particular attention is paid to the case of pure units. We associate to each pure unit a transform called Plancherel transform. We characterize the pure units with the use of their Plancherel transforms. We develop new methods giving a new proof to the well known theorem of Bargmann (see [3]) in the case of Lorentz groups, the Harish-Chandra theorem (see [5]) in the case of semi-simple Lie groups and a well known theorem of Duflo and Moore (see [4]) in the case of general nonunimodular locally compact groups. Our methods allow us to give an explicit expression of the formal operator introduced in [4] by Duflo and Moore.

The Harish-Chandra theorem for the quantum algebrau q (sl (3))

A basis of a quantum universal enveloping algebraU is constructed; the following theorem is proved with the help of this basis: For any nonzero element Μ ∃U, there exists a finite-dimensional representation π such thatπ(u) ≠ 0.

Characters of Non-Connected, Reductive p-Abic Groups

In this paper, we extend to non-connected, reductive groups over p-adic field of characteristic zero Harish-Chandra's theorem on the local integrability of characters.Harish-Chandra's theorem states that the distribution character of an admissible, irreducible representation of a (connected) reductive p-adic group is locally integrable. We show that this extends to any reductive group; just as in the connected case, one even gets a very precise control over the singularities of the character along the singular elements.As will be seen, the proof in the non-connected case is an easy extension of Harish-Chandra's. The reader may wonder why we have bothered to write its generalization completely. The reason is that the original article [8] does not contain proofs for the crucial lemmas, and this makes it impossible to explain why the theorem extends. Because this result is needed for work of Arthur and the author on base change, it has been thought necessary to give complete arguments.

On nilpotency in the Ado-Harish-Chandra theorem on Lie algebra representations

Let L L be a finite-dimensional Lie algebra, over an arbitrary field, and regard L L as embedded in its enveloping algebra UL. Theorem. If K K is an ideal of L L and K K is nilpotent of class q q , then for any r r there exists a finite-dimensional representation ρ \rho of L L which vanishes on all products (in UL) of ≥ q r + 1 \geq qr + 1 elements of K K and is faithful on the subspace of UL spanned by all products of ≤ r \leq r elements of L L . This result sharpens (with respect to nilpotency) the Ado-Iwasawa theorem on the existence of faithful representations and the Harish-Chandra theorem on the existence of representations separating points of UL.

On an Infinitesimal Characterization of the Discrete Series

The aim of this paper is to obtain an infinitesimal characterization of the discrete series of representations of a semisimple Lie group. Let G be a semisimple Lie group, connected and having a finite center. For simplicity we assume that G is contained as a real form of a complex simply connected semisimple Lie group G,. Let K c G be a maximal compact subgroup of G. We assume that rk (G) = rk (K), rk denoting the rank, and fix a maximal torus B c K. Let g be the Lie algebra of G, and f, b, the subalgebras defined by K, B respectively. g, is the complexification of g (g c g,), and (5 is the universal enveloping algebra of ge, while A is the subalgebra of (3 generated by (1, f). A is the set of roots of (g, b), and A, those of (f, b); Pk is a fixed positive system in Ak. L is the lattice of all integral linear functions on bc, i.e., X e bc* such that 2 / e Z for all a e A. L' is the subset of all X e L such that O for all a e A. L, is the set of all X e VC such that 2 / e Z for all f e Ak; C, is the set of all X e b* such that is real and ?0 for all eak. We write L+ = L, n Ca. For any te e Lt, z-t is the unique irreducible St-module with highest weight 4ei (relative to Pk). If se e L n Lt (and only for such ts), z,> gives rise to a K-module, also denoted by zry g = f + 4p is the Cartan decomposition of g. Finally, 3 denotes the center of $ and ?A, the centralizer of K in 6. The elements of L' n Cf parameterize the equivalence classes of the discrete series of representations of G (cf. [6]; see also Section 6). For any A e L' n Cf let co(A) denote the corresponding equivalence class. Fix A e L' n C, and let P be the positive system of roots of (g, b) such that > 0 for all a e P. Define ak, a, by (19) of Section 6. Then A 8k + an e L n Lt and one knows that at least when all the numbers (a e P) are sufficiently large, the discrete class co(AX) contains ZA-3k+an with multiplicity 1 ([8], [9]). By Schur's lemma the elements of D act as scalars on the corresponding isotypical subspace, giving rise to a homomorphism XA of Z into C. A well-known theorem of Harish-Chandra [4] now implies that

Unitary Irreducible Representations of SU(2, 2). II

Paper II of this series [Paper I in J. Math. Phys. 8, 1931 (1967)] is concerned with a general study of the degenerate representations. The explicit expressions for the ``raising'' and ``lowering'' functions ai2(p,q,λ), and bi2(p,q,λ), i = 1, 2, 3, 4 are found. The three Casimir operators C2, C3, and C4 depend on only two complex parameters A and B, a fact reflecting the degenerate nature of the representations under study here. The finite representations are studied first, and thus provide a proof for the degenerate part of Theorem 2, Paper I. The unitary representations are studied next, and we find that there are fourteen classes of degenerate unitary irreducible representations. There are two continuous series, ten discrete series, and two series which depend on one discrete and one continuous parameter. The degenerate part of the D± series is studied, and thus provides an explicit demonstration of Harish-Chandra's theorem.

Unitary Irreducible Representations of SU(2, 2). I

This paper, which is the first of three, is concerned with the general properties of the noncompact group SU(2, 2), the Lie algebra of which is isomorphic to the Dirac algebra. In the course of our study of the unitary representations, we first obtained all the finite dimensional irreducible representations. The associated Young diagrams are shown to have simple properties; the degenerate Young diagrams always denote degenerate representations. Through a theorem of Harish-Chandra, which relates the finite representations to the unitary representations in the discrete series, we are able to obtain explicitly all the unitary infinite-dimensional irreducible representations in this series, both degenerate and non-degenerate. The notion of multiplicity for nondegenerate representations is introduced and discussed in connection with a new operator F3, which is required for a complete labeling of states.