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Representations of parabolic and Borel subgroups. (English) Zbl 1120.20050

Let \(k\) be a locally compact field. Assuming for convenience that the irreducible complex characters of the general linear groups \(\text{GL}_p(k)\) are known, one wants to relate the sets \(\text{Irr}(P)\) of irreducible representations for the various parabolic subgroups \(P\) of all these general linear groups. For a composition \(\lambda=(\lambda_1,\dots,\lambda_s)\) of \(q=\sum_i\lambda_i\) we denote by \(P^\lambda\) the parabolic subgroup of \(\text{GL}_q(k)\) that stabilizes the flag \(k^{\lambda_1}\subseteq k^{\lambda_1+\lambda_2}\cdots\).
By repeated application of Mackey’s theorem on representations of semidirect products, the author expresses the set \(\text{Irr}(P^{(m,n)})\) as a disjoint union of sets \(\text{Irr}(P^\lambda\times\text{GL}_p(k))\), where \(\lambda\) is a composition of \(m\) and \(0\leq p=n-\sum_i(i-1)\lambda_i\). This result can be read in both directions. In particular, taking \(\lambda=(1^n)=(1,\dots,1)\), one learns that the irreducible representations of a Borel subgroup in \(\text{GL}_n(k)\) are determined by representations of maximal parabolic subgroups \(P^{(m,n)}\) in \(\text{GL}_{m+n}(k)\) with \(m=n(n-1)/2\).

MSC:

20G05 Representation theory for linear algebraic groups
22D10 Unitary representations of locally compact groups
20C15 Ordinary representations and characters
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References:

[1] Curtis C. W., Methods of Representation Theory (1990)
[2] Gabriel P., Representations of Finite-Dimensional Algebras (1997)
[3] DOI: 10.1112/plms/s3-52.2.236 · Zbl 0587.20022 · doi:10.1112/plms/s3-52.2.236
[4] DOI: 10.2307/1969423 · Zbl 0046.11601 · doi:10.2307/1969423
[5] Zelevinsky A. V., Representations of Finite Classical Groups (1981) · Zbl 0465.20009
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