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Transactions of the American Mathematical Society | 1989

Remarks on classical invariant theory

Roger Howe

A uniform formulation, applying to all classical groups simultaneously, of the First Fundamental Theory of Classical Invariant Theory is given in terms of the Weyl algebra. The formulation also allows skew-symmetric as well as symmetric variables. Examples illustrate the scope of this formulation.


Journal of the American Mathematical Society | 1989

Transcending classical invariant theory

Roger Howe

Let Sp2, (R) = Sp2, = Sp be the real symplectic group of rank n. Let Sp denote the two-fold cover of Sp. There is a unitary representation, constructed by Shale [Sh] and Weil [WA], of Sp that is of considerable interest in the theory of automorphic forms. We denote this representation by co and call it the oscillator representation. The purpose of this paper is to establish about C? a fact that should be useful for clarifying the structure of spaces of automorphic forms constructed using co. We begin by formulating the result. Let E be a reductive subgroup of Sp, and let E denote the inverse image of E in Sp. Denote by R(E) the set of infinitesimal equivalence classes of continuous irreducible admissible representations of E on locally convex spaces. Let co be the smooth representation of Sp associated to co. Let co be realized on a Hilbert space % and let the subspace of smooth vectors, on which space co is defined, be written . Denote by R9(E , co) the set of elements of (E) which are realized as quotients by co (E)-invariant closed subspaces of tg . Consider a reductive dual pair (G, G) C Sp [H2]. It is not hard to show that G and G commute with one another. Hence, we may regard coI G. G as a representation of Gx G . It is well known [F] that R (Gx G) S9 (G) xR9(G) . The identification associates to p E R(G) and p E i(G) the tensor product p X p. (We note that p X p is not defined as a topological vector space; nevertheless, the infinitesimal equivalence class of p op is well defined.) Select p X p E (G x G) . Suppose that, in fact, p? p E 9(G_ G, c) . Then clearly p E R (G, c) and p E ,9(G, c) . Hence, ( G , co) defines the graph of a correspondence between certain subsets of R9(G, cv) and ?J(G, co) . In fact, the situation is quite precise.


Bulletin of the American Mathematical Society | 1980

On the role of the Heisenberg group in harmonic analysis

Roger Howe

In this article I want to popularize the Heisenberg group, which is remarkably little known considering its ubiquity. I use the word ubiquity advisedly. To justify it, let me give a sample of the many apparently diverse topics where the Heisenberg group reveals itself as an important factor. (1) Representation Theory of Nilpotent Lie Groups (2) Foundations of Abelian Harmonic Analysis (3) Moduli of Abelian Varieties (4) Structure Theory of Finite Groups (5) Theory of Partial Differential Equations (6) Quantum Mechanics (7) Homological Algebra (8) Ergodic Theory (9) Representation Theory of Reductive Algebraic groups (10) Classical Invariant Theory This list could easily be lengthened both by adding new topics and making these more specific, for sometimes the applications are multiple. In fact, one of the most important areas of application I have not mentioned at all, to avoid name-dropping. Why has an object with such wide application gone relatively unnoticed until recently? One can only speculate. One reason might be that the role of the Heisenberg group in many situations is relatively subtle. An investigator might be able to get what he wanted out of a situation while overlooking the extra structure imposed by the Heisenberg group, structure which might enable him to get much more. Such may have been the case with Hermann Weyl, one of the pioneers in introducing the Heisenberg group into Quantum Mechanics [Wyl]. Indeed, many physicists still call the Heisenberg group the Weyl group. When Weyl wrote his book The classical groups, [Wy2] he overlooked the natural occurrence of the Heisenberg group, exploitation of which yields results which one feels Weyl would have liked very much. Again, I gather that an appreciation of the role of the Heisenberg group in rigidifying abelian varieties was an important aspect of Mumfords [Mm] fundamental contributions to their study. Another obstacle to the appreciation of the common underlying structure may have been the very diversity of the topics I listed above, for detection of its presence in one place need not suggest its presence elsewhere. Indeed, investigators in one field may very well never have been aware that the Heisenberg group had been found in some field not seemingly related to theirs. Another factor certainly contrib-


Transactions of the American Mathematical Society | 1973

On the character of Weil’s representation

Roger Howe

The importance of certain representations of symplectic groups, usually called Weil representations, for the general problem of finding representations of certain group extensions is made explicit. Some properties of the character of Weils representation for a finite symplectic group are given and discussed, again in the context of finding representations of group extensions. As a by-product, the structure of anisotropic tori in symplectic groups is given. I. In the title above a pun is intended, for this paper is concerned with two aspects of the celebrated Weil representation-first with its character in the sense of character theory in group representations, then with its character in the more everyday sense of its nature. I think also that the character (in the technical sense) of the Weil representation says something about the character in the general sense. In any case, both facts presented here seem to me rather striking. We proceed to describe them. Let F be a field of characteristic not 2, and let J{(F) be the group of (n + 2) x (n + 2) matrices of the form /1 X1 . .x X Z XI xn 1 1 Q Yi


Transactions of the American Mathematical Society | 1990

Erratum to: “Remarks on classical invariant theory”

Roger Howe

Although this paper circulated as a preprint for 12 years, an error in the argument for the Capelli identity went unnoticed until Professor M. Wakayama read the paper in order to review it. I thank him for his care and for bringing this error to my attention. The error occurs on page 566. The result at issue, that det II is central in /(gyn), is true (and classical) and the general line of argument is sufficient to prove it, but the details of the computation are described incorrectly. The identity on line 3, page 566 is correct, but the description of the calculation drawn from it is not. Instead of terms of the form


arXiv: Representation Theory | 2017

Small Representations of Finite Classical Groups

Shamgar Gurevich; Roger Howe

Finite group theorists have established many formulas that express interesting properties of a finite group in terms of sums of characters of the group. An obstacle to applying these formulas is lack of control over the dimensions of representations of the group. In particular, the representations of small dimensions tend to contribute the largest terms to these sums, so a systematic knowledge of these small representations could lead to proofs of important conjectures which are currently out of reach. Despite the classification by Lusztig of the irreducible representations of finite groups of Lie type, it seems that this aspect remains obscure. In this note we develop a language which seems to be adequate for the description of the “small” representations of finite classical groups and puts in the forefront the notion of rank of a representation. We describe a method, the “eta correspondence”, to construct small representations, and we conjecture that our construction is exhaustive. We also give a strong estimate on the dimension of small representations in terms of their rank. For the sake of clarity, in this note we describe in detail only the case of the finite symplectic groups.


The Schur lectures (1992) (Tel Aviv) | 1995

Perspectives on invariant theory : Schur duality, multiplicity-free actions and beyond

Roger Howe


Archive | 1977

Series and invariant theory

Roger Howe


Archive | 1992

Non-abelian harmonic analysis

Roger Howe; Eng-Chye Tan


Archive | 2010

On A Notion of Rank for Unitary Representations of the Classical Groups

Roger Howe

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Shamgar Gurevich

University of Wisconsin-Madison

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Stephen S. Gelbart

Weizmann Institute of Science

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Eng-Chye Tan

National University of Singapore

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