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The trivial and Steinberg splitting on P^1(F_q)

Example

Assume the Axiom of Choice, used through Tits deformation. For $G = {GL}_{2} (F_{q})$ the spherical principal series is the permutation module $C [P^{1} (F_{q})] = C [G / B]$ of dimension $q + 1$ , and it decomposes as $C [P^{1} (F_{q})] = 1 \oplus St,$ where $1$ is the trivial representation and $St$ is the Steinberg representation of dimension $q$ ; both occur with multiplicity $1$ . Choose the noncanonical hook-length parametrisation so the Hecke character $T_{s} \mapsto q$ matches the trivial $S_{2}$ character and $T_{s} \mapsto - 1$ the sign character. Under this parametrisation of The constituents of the spherical principal series of GL_n the trivial representation corresponds to $λ = (2)$ ( $f^{(2)} = 1$ ) and $St$ to $λ = (1, 1)$ ( $f^{(1, 1)} = 1$ ). The standard Hecke generator $T_{s}$ acts on $1$ by the scalar $q$ and on $St$ by the scalar $- 1$ , matching the two simple $H$ -modules of The two-dimensional Hecke algebra for GL_2(F_q).

Facts & Assumptions

Given: A prime power $q$ , the group $G = {GL}_{2} (F_{q})$ with Borel $B$ and nontrivial Weyl element $s$ , the flag variety $P^{1} (F_{q}) = G / B$ , the spherical principal series $I (1)$ and the finite Hecke algebra $H = e_{B} C [G] e_{B}$ .

[F1]

$I (1) ≅ C [G / B] = C [P^{1} (F_{q})]$ as $C [G]$ -modules, of dimension $q + 1$ (The spherical principal series is the flag permutation module).

[F2]

For the equal-coordinate character with $a = 1$ one has $I (1) ≅ 1 \oplus St$ with $\dim St = q$ and each constituent of multiplicity one; the standard intertwiner $B_{s}$ acts by the scalar $q$ on the one-dimensional constituent and by $- 1$ on $St$ (The equal-coordinate rank-one principal series of GL_2).

[F3]

The constituents of the spherical principal series are indexed by the partitions $λ ⊢ n$ with multiplicities $f^{λ}$ (The constituents of the spherical principal series of GL_n).

[F4]

For $n = 2$ the partitions are $(2)$ and $(1, 1)$ , and the hook-length formula gives $f^{(2)} = f^{(1, 1)} = 1$ (The hook length formula).

[F5]

The two-dimensional Hecke algebra $H$ of ${GL}_{2} (F_{q})$ is isomorphic to $C \oplus C$ with two simple modules, and the generator $T_{s}$ acts by $q$ on one and by $- 1$ on the other (The two-dimensional Hecke algebra for GL_2(F_q)).

[F6]

Assume AC; the Tits-deformation isomorphism identifies the simple $H$ -modules with those of $C [S_{2}]$ , so the partition labels above are attached through the noncanonical isomorphism (The Axiom of Choice, The finite Hecke algebra is non-canonically isomorphic to the group algebra of S_n).

Proof

technique · direct

1.1F1F2

By [F1] the module $C [P^{1} (F_{q})]$ is $I (1)$ of dimension $q + 1$ . By [F2] it splits as $1 \oplus St$ with $\dim St = q$ , both constituents of multiplicity one, and the standard intertwiner acts by $q$ on $1$ and by $- 1$ on $St$ .

2.1F5step 1.1

By [F5] the Hecke algebra $H ≅ C \oplus C$ has exactly two simple modules, and $T_{s}$ acts on them by the scalars $q$ and $- 1$ ; these match the two constituents of step 1.1 through the identification ${End}_{G} (C [G / B]) ≅ H^{o p} ≅ H$ .

3.1F3F4F6step 1.1step 2.1

By [F3] the constituents of the spherical principal series are indexed by the partitions $λ ⊢ 2$ , namely $(2)$ and $(1, 1)$ , and by [F4] both occur with multiplicity $f^{(2)} = f^{(1, 1)} = 1$ , agreeing with the multiplicity-one splitting of step 1.1. Under the noncanonical Tits parametrisation we may choose the matching so that the Hecke character $T_{s} \mapsto q$ labels the trivial constituent $1$ and $T_{s} \mapsto - 1$ labels the sign character, hence $1$ corresponds to $(2)$ and $St$ to $(1, 1)$ .

4.1F6step 1.1step 2.1step 3.1∎

Steps 1.1, 2.1 and 3.1 give the splitting $C [P^{1}] = 1 \oplus St$ with $\dim St = q$ , the multiplicity-one statement, the $T_{s}$ -eigenvalues $q$ and $- 1$ , and the partition labels under the noncanonical parametrisation. AC is carried only from the Tits-deformation supplier [F6], as declared.

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Sources

Masao Oi, Representation Theory of Finite Groups of Lie Type - Proposition 2.8 and its proof, printed pp. 12-13 (standard reference, not scraped)
Jay Taylor, Finite Reductive Groups - Example 5.22, printed p. 46 (standard reference, not scraped)
Ivan Losev, Lecture 8: Representations of GL_n(F_q) - Corollary 2.4 and Theorem 2.5 for $n=2$, PDF pp. 4-5 (standard reference, not scraped)