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The constituents of the spherical principal series of GL_n

Statement

Assume the Axiom of Choice, used through Tits deformation. Let $G = {GL}_{n} (F_{q})$ and let $C [G / B]$ be the permutation module on the complete flags. The set of isomorphism classes of simple constituents of $C [G / B]$ is in bijection with the set ${λ : λ ⊢ n}$ of partitions of $n$ : the bijection is the composite of the endomorphism-algebra parametrisation of Constituent multiplicities are dimensions of simple modules over the endomorphism algebra, the Tits isomorphism ${End}_{G} (C [G / B]) ≅ C [S_{n}]$ of The finite Hecke algebra is non-canonically isomorphic to the group algebra of S_n and the classification of the irreducible $C [S_{n}]$ -modules by partitions. Write $V_{λ}$ for the simple $C [G]$ -module attached to $λ$ by that bijection. Then $C [G / B] ≅ ⨁_{λ ⊢ n} V_{λ}^{\oplus f^{λ}},$ where $f^{λ} = \dim_{C} S^{λ}$ is the number of standard $λ$ -tableaux, and the multiplicities satisfy $\dim_{C} {Hom}_{G} (V_{λ}, C [G / B]) = f^{λ}, {End}_{G} (C [G / B]) ≅ \prod_{λ ⊢ n} M_{f^{λ}} (C) .$ Equivalently, the constituents of the spherical principal series $I (1)$ are indexed by the partitions of $n$ and the constituent indexed by $λ$ occurs with multiplicity $f^{λ}$ .

Facts & Assumptions

Given: $G = {GL}_{n} (F_{q})$ with Borel $B$ , the permutation module $M : = C [G / B]$ on the complete flags, the spherical principal series $I (1)$ , the finite Hecke algebra $H = e_{B} C [G] e_{B}$ and the symmetric group $S_{n}$ with its partition-indexed Specht modules $S^{λ}$ .

[F1]

$I (1) ≅ C [G / B]$ as $C [G]$ -modules (The spherical principal series is the flag permutation module).

[F2]

Right multiplication identifies $H ≅ {End}_{C [G]} (C [G] e_{B})^{o p} ≅ {End}_{G} (C [G / B])^{o p}$ , and $H ≅ H^{o p}$ (The finite Hecke algebra as a convolution corner and its endomorphism interpretation).

[F3]

Assume AC; the Tits-deformation isomorphism gives $H ≅ C [S_{n}]$ , preserving the number and dimensions of simple modules (The finite Hecke algebra is non-canonically isomorphic to the group algebra of S_n, The Axiom of Choice).

[F4]

For a finite-dimensional semisimple $C$ -algebra $A$ and a finite-dimensional semisimple $A$ -module $M$ with $M ≅ ⨁_{i} V_{i}^{\oplus m_{i}}$ over pairwise non-isomorphic simples $V_{i}$ , the algebra $E = {End}_{A} (M)$ is semisimple with $E ≅ \prod_{i} M_{m_{i}} (C)$ ; the simple $E$ -modules are the spaces ${Hom}_{A} (V_{i}, M)$ of dimension $m_{i}$ , and they form a complete set of simple isomorphism classes (Constituent multiplicities are dimensions of simple modules over the endomorphism algebra).

[F5]

The simple $C [S_{n}]$ -modules are exactly the Specht modules $S^{λ}$ , $λ ⊢ n$ , pairwise non-isomorphic (Specht modules classify the complex irreducibles of $S_{n}$ , Partitions, English diagrams, and conjugation).

[F6]

$\dim_{C} S^{λ} = f^{λ}$ , the number of standard $λ$ -tableaux, by the hook-length formula (The hook length formula).

[F7]

Every finite-dimensional complex representation of the finite group $G$ is semisimple, since $char C = 0 ∤ ∣ G ∣$ (Maschke's theorem for finite groups over fields whose characteristic does not divide $∣ G ∣$ ).

Proof

technique · direct

1.1F1F2F4F7

By [F7] the $C [G]$ -module $M = C [G / B]$ is semisimple, and by [F1] it is the spherical principal series $I (1)$ . Put $E : = {End}_{G} (M)$ ; by [F2] $E ≅ H^{o p} ≅ H$ , a finite-dimensional semisimple algebra by the finite Hecke algebra theorem. Applying [F4] to $M$ over $A = C [G]$ , the simple constituents $V_{i}$ of $M$ are in bijection with the simple $E$ -modules ${Hom}_{G} (V_{i}, M)$ , each of dimension equal to the multiplicity $m_{i}$ of $V_{i}$ in $M$ .

2.1F3F5F6step 1.1

By [F3] there is an isomorphism $E ≅ C [S_{n}]$ preserving the number and dimensions of simple modules. By [F5] the simple $C [S_{n}]$ -modules are the Specht modules $S^{λ}$ , $λ ⊢ n$ , and by [F6] $\dim_{C} S^{λ} = f^{λ}$ . Transporting along $E ≅ C [S_{n}]$ , the simple constituents of $M$ are therefore indexed by the partitions $λ ⊢ n$ : define $V_{λ}$ as the constituent corresponding to $S^{λ}$ under the transport. Its multiplicity in $M$ equals $\dim_{C} S^{λ} = f^{λ}$ by step 1.1, and $\dim_{C} {Hom}_{G} (V_{λ}, M) = f^{λ}$ .

3.1F4step 1.1step 2.1

Assembling steps 1.1 and 2.1, $M$ is the direct sum of its constituents with multiplicities $f^{λ}$ , that is $M ≅ ⨁_{λ ⊢ n} V_{λ}^{\oplus f^{λ}}$ ; by [F4] the endomorphism algebra is $E ≅ \prod_{λ ⊢ n} M_{f^{λ}} (C)$ , agreeing with the transport in step 2.1. This is the stated description of the constituents of the spherical principal series.

4.1F3F4step 1.1step 2.1step 3.1∎

Steps 1.1, 2.1 and 3.1 give the bijection, the multiplicities and the endomorphism algebra. AC is carried only from the Tits-deformation supplier [F3], as declared; the remaining arguments use Maschke's theorem and finite-dimensional semisimple module theory over $C$ .

Depends on

Used by

The constituents of a general finite principal series Corollary
Regular and singular torus characters in GL₃(F_q) Example
The q=2 torus boundary Example
The trivial and Steinberg splitting on P¹(F_q) Example

Dependency tree · two levels

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