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The constituents of a general finite principal series

Statement

Assume the Axiom of Choice, used through Tits deformation. Let $G = {GL}_{n} (F_{q})$ , $χ \in \hat{T}$ with equal-character blocks of sizes $n_{1}, \dots, n_{k}$ and Weyl stabiliser $W_{χ} = S_{n_{1}} \times \dots \times S_{n_{k}}$ . Then the isomorphism classes of simple constituents of the principal series $I (χ)$ are indexed by the tuples $(λ^{(1)}, \dots, λ^{(k)})$ of partitions $λ^{(r)} ⊢ n_{r}$ , and the multiplicity of the constituent attached to such a tuple equals $\prod_{r = 1}^{k} f^{λ^{(r)}},$ the product of the numbers of standard tableaux of the parts; equivalently ${End}_{G} (I (χ)) ≅ \prod_{(λ^{(1)}, \dots, λ^{(k)})} M_{\prod_{r} f^{λ^{(r)}}} (C) .$ In particular, if $χ$ is regular ( $k = n$ , all $n_{r} = 1$ ) then $I (χ)$ is irreducible, and if $χ$ is trivial ( $k = 1$ , $n_{1} = n$ ) the multiplicities are the hook-length numbers $f^{λ}$ of The constituents of the spherical principal series of GL_n.

Facts & Assumptions

Given: $G = {GL}_{n} (F_{q})$ with Borel $B$ and diagonal torus $T$ , a character $χ \in \hat{T}$ with equal-character block sizes $n_{1}, \dots, n_{k}$ , the stabiliser $W_{χ} = S_{n_{1}} \times \dots \times S_{n_{k}}$ , the principal series $I (χ)$ and its endomorphism algebra $E = {End}_{G} (I (χ))$ .

[F1]

Assume AC; $E ≅ ⨂_{r} H_{q} (S_{n_{r}}) = H_{q} (W_{χ})$ , and $E ≅ C [W_{χ}]$ preserving the number and dimensions of simple modules, with the identification $E ≅ H_{q} (W_{χ})$ itself choice-free (The endomorphism algebra of a general finite principal series, The Axiom of Choice).

[F2]

$C [G]$ and hence every finite-dimensional complex $G$ -module is semisimple (Maschke's theorem for finite groups over fields whose characteristic does not divide $∣ G ∣$ ).

[F3]

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

[F4]

A nonzero semisimple finite-dimensional $C$ -algebra is isomorphic to a product of matrix algebras, with simple modules the natural column modules of the factors (The trace form detects semisimplicity over the complex numbers).

[F5]

The simple $C [S_{m}]$ -modules are the Specht modules $S^{λ}$ , $λ ⊢ m$ , pairwise non-isomorphic, and $\dim_{C} S^{λ} = f^{λ}$ is the number of standard $λ$ -tableaux (Specht modules classify the complex irreducibles of $S_{n}$ , The hook length formula).

Proof

technique · direct

1.1F1F2F3

By [F2] the module $I (χ)$ is a semisimple $C [G]$ -module, so [F3] applies to it with $A = C [G]$ and $M = I (χ)$ : the constituents $V_{i}$ of $I (χ)$ are in bijection with the simple $E$ -modules ${Hom}_{G} (V_{i}, I (χ))$ , and the multiplicity of $V_{i}$ equals the dimension of that simple $E$ -module. By [F1] we have $E ≅ ⨂_{r} H_{q} (S_{n_{r}})$ and, through the Tits isomorphism, $E ≅ C [W_{χ}] = ⨂_{r} C [S_{n_{r}}]$ .

2.1F4F5step 1.1algebra

By [F4] each factor is a product of matrix algebras, $C [S_{n_{r}}] ≅ \prod_{i} M_{d_{i}^{(r)}} (C)$ , with simple modules the column modules of dimensions $d_{i}^{(r)}$ ; by [F5] these simple modules are exactly the Specht modules $S^{λ^{(r)}}$ , $λ^{(r)} ⊢ n_{r}$ , of dimension $f^{λ^{(r)}}$ . For matrix algebras there is an algebra isomorphism $M_{a} (C) \otimes M_{b} (C) ≅ M_{a b} (C)$ sending the matrix units $E_{i j} \otimes F_{k l}$ to the matrix units indexed by the pairs $(i, k), (j, l)$ , which is multiplicative because the products of pairs multiply componentwise; tensoring over the factors therefore gives $⨂_{r} C [S_{n_{r}}] ≅ \prod_{(λ^{(1)}, \dots, λ^{(k)})} M_{\prod_{r} f^{λ^{(r)}}} (C)$ , with the simple module of the tuple $(λ^{(1)}, \dots, λ^{(k)})$ being the external tensor product $S^{λ^{(1)}} ⊠ \dots ⊠ S^{λ^{(k)}}$ of dimension $\prod_{r} f^{λ^{(r)}}$ .

3.1F3step 2.1algebra

Transporting the simple $E$ -modules of step 2.1 along the isomorphism $E ≅ C [W_{χ}]$ and applying the parametrisation of step 1.1, the constituents of $I (χ)$ are indexed by the tuples $(λ^{(1)}, \dots, λ^{(k)})$ , and the constituent attached to a tuple has multiplicity $\prod_{r} f^{λ^{(r)}}$ ; by [F3] the endomorphism algebra is $\prod_{(λ^{(1)}, \dots, λ^{(k)})} M_{\prod_{r} f^{λ^{(r)}}} (C)$ , as displayed. If $χ$ is regular then every $n_{r} = 1$ and the only partition of each $n_{r} = 1$ is $(1)$ , with $f^{(1)} = 1$ , so the tuple index set has one element and its multiplicity is $1$ : $I (χ)$ is irreducible. If $χ$ is trivial then $k = 1$ , $n_{1} = n$ , and the formula is the spherical multiplicity formula of The constituents of the spherical principal series of GL_n.

4.1F1F3step 1.1step 2.1step 3.1∎

Steps 1.1, 2.1 and 3.1 give the index set, the multiplicities, the endomorphism algebra and the two boundary cases. AC is carried only from the Tits-deformation supplier inside [F1], as declared; all remaining arguments are finite-dimensional over $C$ .

Depends on

Used by

Regular and singular torus characters in GL₃(F_q) Example
The Tits isomorphism is noncanonical Remark

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Sources

Olivier Dudas and Jean Michel, Lectures on Finite Reductive Groups and Their Representations - Corollary 11.12 and Example 11.13 (parametrisation of the series by irreducibles of $W(L,N)^F$), printed p. 50 (standard reference, not scraped)
Jay Taylor, Finite Reductive Groups - Theorem 5.21 and Example 5.22 (multiplicities are dimensions of simple modules of $W^F\cong S_n$), printed pp. 45-46 (standard reference, not scraped)
Ivan Losev, Lecture 8: Representations of GL_n(F_q) - Section 2.1 (summands indexed by simple modules of the endomorphism algebra), PDF pp. 3-4 (standard reference, not scraped)