Representations of generic algebras and finite groups of Lie type.

*(English)*Zbl 0537.20018Let G be the group of \({\mathbb{F}}_ q\)-rational points of a connected reductive \({\mathbb{F}}_ q\)-group G, where \({\mathbb{F}}_ q\) is the Galois field of q elements. The group G has a split (B,N)-pair in characteristic p \((q=p^ e)\). Each standard parabolic subgroup \(P_ J\) of G (where J is a subset of the simple roots associated with the (B,N)-structure) has a standard Levi decomposition \(P_ J=M_ JU_ J\), where \(U_ J\) is the ”unipotent radical” of \(P_ J\), and \(M_ J\) the standard Levi component of \(P_ J.\)

From the introduction: ”The passage between the representation theory of G and of its Weyl group W is accomplished by means of certain ”generic algebras”, which are associative algebras over a complex polynomial ring and which ”specialize” to various complex algebras which have significance for the representation theory of G and W. More precisely, the ”Harish-Chandra” principle (§ 3) organizes the representations of G into ”series” which consist of the irreducible constituents of a fixed induced cuspidal representation \(Ind(P_ L\to G;\tilde D)\) (where \(P_ L\) is a parabolic subgroup of G and D is a cuspidal representation of \(M_ L)\). In [Invent. Math. 58, 37-64 (1980; Zbl 0435.20023)], the constituents of a single series where studied by elucidating the structure of the endomorphism algebra of \(Ind(P_ L\to G;\tilde D).\) In § 1 of the present work we make explicit the connection between the decomposition of induced representations and the representation theory of the associated endomorphism algebras by introducing two pairs of adjoint functors and studying the way they interact. This transfers the study of any particular series to the study of a certain endomorphism algebra.”

The principal result of the paper cited above was that the endomorphism algebras concerned have an explicit presentation in terms of the ramification group W(D) of D and certain powers \(p_ a\) of the characteristic. ”Replacing these \(p_ a\) by indeterminates \(u_ a\), one constructs ”generic algebras” over the ring \({\mathbb{C}}[\{u_ a\}]\); § 4 of the present work is devoted to the study of these generic algebras and of the connection between their representation theory and that of the ramification groups W(D). This provides (Theorem (4.8)) explicit descriptions of each of the series of representations of G, in terms of data relating to the cuspidal representation and its ramification group. In § 5 a refinement (5.6) of this description is proved, in which G is replaced by the Levi component \(M_ J\) of the parabolic subgroup \(P_ J.''\)

”In general terms, the Comparison Theorem (5.9) states that if \(\phi\) and \(\psi\) are characters of appropriate projective representations of the ramification group \(W(\delta)\) and its subgroup \(W^ w_ J\cap W(\delta)\) (where \(wL\subset J\subset \Pi\) and \(\delta\) is an irreducible cuspidal character of \(M_ L)\), and \(\zeta_{\phi}\), \(\eta_{w\psi}\) are the corresponding characters in the \(\delta\)-series of G and \(w\delta\)-series of \(M_ J\) respectively, then the multiplicities of \(\zeta_{\phi}\) in \(Ind(P_ J\to G;{\tilde \eta}_{w\psi})\) and of \(\phi\) in \(Ind(W^ w_ J\cap W(\delta)\to W(\delta);\psi)\) are equal. The proof is accomplished by showing that both multiplicities are equal to a certain multiplicity which arises in the generic algebra context.”

As an application of this ”comparison theorem” the authors are able to determine precisely what the dual of an arbitrary irreducible character of G is, thus generalizing the results of C. W. Curtis [J. Algebra 62, 320-332 (1980; Zbl 0426.20006)].

From the introduction: ”The passage between the representation theory of G and of its Weyl group W is accomplished by means of certain ”generic algebras”, which are associative algebras over a complex polynomial ring and which ”specialize” to various complex algebras which have significance for the representation theory of G and W. More precisely, the ”Harish-Chandra” principle (§ 3) organizes the representations of G into ”series” which consist of the irreducible constituents of a fixed induced cuspidal representation \(Ind(P_ L\to G;\tilde D)\) (where \(P_ L\) is a parabolic subgroup of G and D is a cuspidal representation of \(M_ L)\). In [Invent. Math. 58, 37-64 (1980; Zbl 0435.20023)], the constituents of a single series where studied by elucidating the structure of the endomorphism algebra of \(Ind(P_ L\to G;\tilde D).\) In § 1 of the present work we make explicit the connection between the decomposition of induced representations and the representation theory of the associated endomorphism algebras by introducing two pairs of adjoint functors and studying the way they interact. This transfers the study of any particular series to the study of a certain endomorphism algebra.”

The principal result of the paper cited above was that the endomorphism algebras concerned have an explicit presentation in terms of the ramification group W(D) of D and certain powers \(p_ a\) of the characteristic. ”Replacing these \(p_ a\) by indeterminates \(u_ a\), one constructs ”generic algebras” over the ring \({\mathbb{C}}[\{u_ a\}]\); § 4 of the present work is devoted to the study of these generic algebras and of the connection between their representation theory and that of the ramification groups W(D). This provides (Theorem (4.8)) explicit descriptions of each of the series of representations of G, in terms of data relating to the cuspidal representation and its ramification group. In § 5 a refinement (5.6) of this description is proved, in which G is replaced by the Levi component \(M_ J\) of the parabolic subgroup \(P_ J.''\)

”In general terms, the Comparison Theorem (5.9) states that if \(\phi\) and \(\psi\) are characters of appropriate projective representations of the ramification group \(W(\delta)\) and its subgroup \(W^ w_ J\cap W(\delta)\) (where \(wL\subset J\subset \Pi\) and \(\delta\) is an irreducible cuspidal character of \(M_ L)\), and \(\zeta_{\phi}\), \(\eta_{w\psi}\) are the corresponding characters in the \(\delta\)-series of G and \(w\delta\)-series of \(M_ J\) respectively, then the multiplicities of \(\zeta_{\phi}\) in \(Ind(P_ J\to G;{\tilde \eta}_{w\psi})\) and of \(\phi\) in \(Ind(W^ w_ J\cap W(\delta)\to W(\delta);\psi)\) are equal. The proof is accomplished by showing that both multiplicities are equal to a certain multiplicity which arises in the generic algebra context.”

As an application of this ”comparison theorem” the authors are able to determine precisely what the dual of an arbitrary irreducible character of G is, thus generalizing the results of C. W. Curtis [J. Algebra 62, 320-332 (1980; Zbl 0426.20006)].

Reviewer: G.Noskov

##### MSC:

20G05 | Representation theory for linear algebraic groups |

20G40 | Linear algebraic groups over finite fields |

20C30 | Representations of finite symmetric groups |

##### Keywords:

split (B,N)-pair; standard parabolic subgroup; simple roots; Levi decomposition; Weyl group; decomposition of induced representations; endomorphism algebras; ramification group; generic algebras; series of representations; Comparison Theorem; projective representations; irreducible cuspidal character; multiplicities; irreducible character
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\textit{R. B. Howlett} and \textit{G. I. Lehrer}, Trans. Am. Math. Soc. 280, 753--779 (1983; Zbl 0537.20018)

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##### References:

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