Bilinear and Quadratic Forms on Rational Modules of Split Reductive Groups

Quasi-hereditary algebras were introduced by L. Scott [S] in order to study highest weight categories arising in the representation theory of semisimple complex Lie algebras and algebraic groups, and important results were proved by Cline, Parshall and Scott (see [CPS1,2]). These algebras can be defined entirely in ring-theoretic terms; and they were studied from this point of view by Dlab and Ringel (see [DR1,2], [R1,2]) and by others. In particular it turns out that quasi-hereditary algebras are quite common.

In this lecture we indicate how to complete the last step in the process outlined at the beginning of Part II: to take our knowledge of the classification and representation theory of complex algebras and groups and deduce the corresponding statements in the real case. We do this in the first section, giving a list of the simple classical real Lie algebras and saying a few words about the corresponding groups and their (complex) representations. The existence of a compact group whose Lie algebra has as complexification a given semisimple complex Lie algebra makes it possible to give another (indeed, the original) way to prove the Weyl character formula; we sketch this in §26.2. Finally, we can ask in regard to real Lie groupsGa question analogous to one asked for the representations of fmite groups in §3.5: which of the complex representationsVofGactually come from real ones. We answer this in the most commonly encountered cases in §26.3. In this final lecture, proofs, when we attempt them, are generally only sketched and may require more than the usual fortitude from the reader.KeywordsIrreducible RepresentationSpin RepresentationReal FormCartan SubalgebraExterior PowerThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

If K is a connected compact Lie group with simple Lie algebra and if k is an integer relatively prime to the order of the Weyl group W of K then the number ν ( K , k ) \nu (K,k) of conjugacy classes of K consisting of elements x satisfying x k = 1 {x^k} = 1 is given by \[ ν ( K , k ) = ∏ i = 1 l m i + k m i + 1 , \nu (K,k) = \prod \limits _{i = 1}^l {\frac {{{m_i} + k}}{{{m_i} + 1}},} \] where l is the rank of K and m 1 , … , m l {m_1}, \ldots ,{m_l} are the exponents of W. If G is the complexification of K then we have ν ( G , k ) = ν ( K , k ) \nu (G,k) = \nu (K,k) without any restriction on k.

Let A be a finite-dimensional algebra which is quasi-hereditary with re- spect to the poset (�,�), with standard modules �(�) for 2 �. Let F(�) be the category of A-modules which have filtrations where the quotients are standard modules. We determine some inductive results on the relative Auslander-Reiten quiver of F(�). Quasi-hereditary algebras were introduced by Cline, Parshall and Scott in the study of highest weight categories arising in the representation theory of semisimple Lie algebras and algebraic groups (2), and they were studied by Dlab and Ringel and others (4, 5). LetA be quasi-hereditary. Then the simple A-modules are labelled as L(�) forin some partially ordered set (�,≤). Of central importance are the standard modules, denoted by �(�), and the categoryF(�) of modules which have a filtration where the quotients are standard modules. In (12) it was proved that this category has relative Auslander-Reiten (AR) sequences. The aim of our paper is to present some inductive results on the relative Auslander-Reiten quiver ofF(�). For any subset of �, letF (�) be the category of modules inF(�) where all quotients are �(�) with �∈ . If � is an ideal in the poset (�,≤) then there is a subalgebra A of A which is quasi-hereditary with poset (, ≤). In this case, the categoryF (�) is equivalent toF(�A ). A main result in this paper relates, for a non-projective indecomposable mod- ule X inF (�), the two relative Auslander-Reiten sequences with cokernel X inF(�) and inF (�) (Theorem 2.5). By Ringel duality, there is an equivalent version for categories of modules filtered by costandard mod- ules. We describe this and some variations in Section 3. We continue with further results on relative Auslander-Reiten sequences. In particular, we determine certain relative Auslander-Reiten sequences where a projective- injective summand occurs; this generalizes the usual almost split sequence where a projective-injective module occurs.

Symmetry reduction for the Kadomtsev-Petviashvili equation using its infinite dimensional invariance group

The first part of this chapter treats Lie algebras, beginning with definitions and many examples. The notions of solvable, nilpotent, radical, semisimple, and simple are introduced, and these notions are followed by a discussion of the effect of a change of the underlying field. The idea of a semidirect product begins the development of the main structural theorems for real Lie algebras—the iterated construction of all solvable Lie algebras from derivations and semidirect products, Lie's Theorem for solvable Lie algebras, Engel's Theorem in connection with nilpotent Lie algebras, and Cartan's criteria for solvability and semisimplicity in terms of the Killing form. From Cartan's Criterion for Semisimplicity, it follows that semisimple Lie algebras are direct sums of simple Lie algebras. Cartan's Criterion for Semisimplicity is used also to provide a long list of classical examples of semisimple Lie algebras. Some of these examples are defined in terms of quaternion matrices. Quaternion matrices of size n-by-n may be related to complex matrices of size 2n-by-2n. The treatment of Lie algebras concludes with a study of the finite-dimensional complex-linear representations of sl[(2, ℂ). There is a classification theorem for the irreducible representations of this kind, and the general representations are direct sums of irreducible ones. Section 10 contains a review of the elementary theory of Lie groups and their Lie algebras. The abstract theory as in Chevalley [1946] is summarized, and the correspondence is made with the concrete theory of closed linear groups, where the Lie algebra is obtained as the space of derivatives at t = 0 of smooth curves in the group passing through the identity at t = 0. The section ends with a discussion of the adjoint representation. The remainder of the chapter explores some aspects of the connection between Lie groups and Lie algebras. One aspect is the relationship between automorphisms and derivations. The derivations of a semisimple Lie algebra are inner, and consequently the identity component of the group of automorphisms of a semisimple Lie algebra consists of inner automorphisms. In addition, simply connected solvable Lie groups may be built one dimension at a time as semidirect products with ℝ1, and consequently they are diffeomorphic to Euclidean space. For simply connected nilpotent groups the exponential map is itself a diffeomorphism. The earlier long list of classical semisimple Lie algebras corresponds to a list of the classical semisimple Lie groups. The issue that needs attention for these groups is their connectedness, and this is proved by using the polar decomposition of matrices.

We give a general theory of matrix elements (ME’s) of the unitary irreducible representations (UIR’s) of linear semisimple Lie groups and of reductive Lie groups. This theory connects together the following things, (1) MEUIR’s of all the representation series of a noncompact Lie group, (2) MEUIR’s of compact and noncompact forms of the same complex Lie group. The theory presented is based on the results of the theory of the principal nonunitary series representations and on a theorem which states that ME’s of the principal nonunitary series representations are entire analytic functions of continuous representation parameters. The principle of analytic continuation of Clebsch–Gordan coefficients (CGC’s) of finite dimensional representations to CGC’s of the tensor product of a finite and an infinite dimensional representation and to CGC’s of the tensor product of two infinite dimensional representations is proved. ME’s for any UIR of the group U(n) and of the group U(n,1) are obtained. The explicit expression for all CGC’s summed over the multiplicity of the irreducible representation in the tensor product decomposition is derived.

Weights for symmetric and general linear groups

An estimation problem in compact Lie groups

Relative BGG sequences; II. BGG machinery and invariant operators

Let G be a semisimple Lie group, g its Lie algebra. The Killing form on g is a nondegenerate symmetric bilinear form which induces an isomorphism a: g* -. g. The Casimir operator is the unique biinvariant differential operator on G which annihilates constants and whose principal symbol is the pullback of the Killing form under the isomorphism a. Let C denote the Casimir operator and C-(G) the space of infinitely differentiable functions on G. THEOREM. If G is a non-compact semisimple Lie group with finite center, f e C-(G), then there is a u e C-(G) such that Cu = f. Since G is not compact, C is not elliptic but ultrahyperbolic. Analogous questions have been considered by S. Helgason [3] and A. Cerezo and F. Rouviere [1]. Helgason proves local solvability of all biinvariant differential operators and Cerezo and Rouviere prove global solvability of the Casimir operator on complex semisimple Lie groups. The latter authors prove that the imaginary part of the complex Casimir operator on SL(2, C) is not globally solvable. This shows that not every biinvariant operator is globally solvable. Henceforward G will denote a group satisfying the hypotheses of the theorem and g will denote the Lie algebra of left invariant vector fields on G. Proof. The remainder of the paper is devoted to a proof of this theorem. In spite of its length the proof is simple in outline. A result from functional analysis and Hormander's propagation of singularities theorem can be combined to show that it suffices to demonstrate three things.

Generalized Serre relations for Lie algebras associated with positive unit forms

In this paper we describe the new model of the representations of the current groups with a semisimple Lie group of the rank one. In the earlier papers of 70-80-th (Araki, Gelfand-Graev-Vershik) had posed the problem about irreducible representations of the current group for $SL(2,R)$, and was used for this the well-known Fock space-structure That construction could be applied to the arbitrary locally compact group,and is based on a so called special representation of the original group $G$, with nontrivial 1-cohomology. A new construction uses the special property of one dimensional extensions (semi-direct product)of the nilpotent groups which allows immediately to produce the special representation of the group and then to apply the quasi-Poisson construction from the previous papers by authors in order to construct the representation of current group. The parabolic subgroup of the semisimple Lie group of rank one has such semidirect product, and special representation of it can be extended onto whole semisimple group. As result we obtain needed new model of the irreducible representation of semi-simple current groups.

Multiplicity One Theorems for Generalized Gelfand-Graev Representations of Semisimple Lie Groups and Whittaker Models for the Discrete Series

This paper is a personal look at some issues in the representation theory of Lie groups having to do with the role of commutative hypergroups, bi-modules, and the construction of representations. We begin by considering Frobenius’ original approach to the character theory of a finite group and extending it to the Lie group setting, and then introduce bi-modules as objects intermediate between characters and representations in the theory. A simplified way of understanding the formalism of geometric quantization, at least for compact Lie groups, is presented, which leads to a canonical bi-module of functions on an integral coadjoint orbit. Some meta-mathematical issues relating to the construction of representations are considered.