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The rank-one Hecke parameter for equal torus characters

Statement

For Weyl-sorted $η$ , each simple $s \in S_{η}$ exchanges two adjacent equal coordinates $a, a$ . The associated rank-one Levi has principal series $(a \circ \det) \oplus (St \otimes a \circ \det)$ on its ${GL}_{2}$ factor, tensored with one-dimensional characters on the remaining torus factors, with constituent degrees $1, q$ . The canonical raw intertwiner has eigenvalues $a (- 1) q$ and $- a (- 1)$ . The normalized $T_{s} = a (- 1)^{- 1} B_{s}$ satisfies $T_{s}^{2} = (q - 1) T_{s} + q i d$ ; thus the parameter is exactly $q$ , and the normalization of Length-additive products of the standard intertwiners gives $λ_{w} = \prod_{r} a_{r} (- 1)^{- ℓ (w_{r})}$ . For arbitrary $χ$ this assertion holds for the transported generators after Weyl sorting. In particular the naive generator fails the parameter- $q$ relation when $a (- 1) = - 1$ : for $q = 3$ and the nontrivial character of $F_{3}^{\times}$ , its eigenvalue on $a \circ \det$ is $- 3$ , which is neither $3$ nor $- 1$ . No choice principle is used.

Facts & Assumptions

Given: A Weyl-sorted character $η = (a_{1}^{n_{1}}, \dots, a_{k}^{n_{k}})$ of the diagonal torus $T$ of $G = {GL}_{n} (F_{q})$ with distinct $a_{r}$ , a simple reflection $s \in W_{η}$ in the block $r$ of size $n_{r} \geq 2$ , the characters $ψ_{r} = a_{r} \circ \det$ and $ρ = \prod_{r} a_{r} \circ \det_{r}$ of $L = \prod_{r} {GL}_{n_{r}} (F_{q})$ , the idempotents $e_{r} = ∣ B_{r} ∣^{- 1} \sum_{b \in B_{r}} {\tilde{η}}_{r} (b)^{- 1} b$ and $e_{B_{r}}$ , and the operators $B_{s} = R_{Θ_{s^{- 1}}}$ , $T_{s} = ρ (\dot{s})^{- 1} B_{s}$ of Standard intertwining operators for the finite principal series and Length-additive products of the standard intertwiners.

[F1]

For the block ${GL}_{n_{r}}$ the standard basis element $T_{s}^{(r)} = q e_{B_{r}} \dot{s} e_{B_{r}}$ satisfies $T_{s}^{(r) 2} = (q - 1) T_{s}^{(r)} + q e_{B_{r}}$ (The rank-one quadratic relation in the finite Hecke algebra, The Bruhat double-coset basis of the finite Hecke algebra).

[F2]

The map $m_{r} (g) = ψ_{r} (g)^{- 1} g$ is an algebra automorphism of $C [{GL}_{n_{r}} (F_{q})]$ with $m_{r} (e_{B_{r}}) = e_{r}$ ; it sends $T_{s}^{(r)} = q e_{B_{r}} \dot{s} e_{B_{r}}$ to $ψ_{r} (\dot{s})^{- 1} Θ_{s}^{(r)}$ with $Θ_{s}^{(r)} = q e_{r} \dot{s} e_{r}$ , because $ψ_{r} (b) = a_{r} (\det b) = {\tilde{η}}_{r} (b)$ for upper triangular $b$ (The group ring $R [G]$ is a unital $R$ -algebra with basis $G$ , and each $g \in G$ is a unit of $R [G]$ , For same-sized finite square matrices over a commutative ring, $\det (A B) = \det (A) \det (B)$ , The determinant of a triangular matrix is the product of its diagonal entries, Length-additive products of the standard intertwiners, The principal series module for finite GL_n).

[F3]

$ψ_{r} (\dot{s}) = a_{r} (\det P_{s}) = a_{r} (- 1) \in {1, - 1}$ , and $ρ (\dot{s}) = a_{r} (- 1)$ for $s$ in block $r$ . The element $Θ_{s}^{(r)}$ satisfies $Θ_{s}^{(r) 2} = (q - 1) ψ_{r} (\dot{s}) Θ_{s}^{(r)} + q e_{r}$ , and $e_{r}$ acts as the identity on the module $e_{r} C [{GL}_{n_{r}}] e_{r}$ . [F2, algebra]

[F4]

For the rank-one Levi $M = {GL}_{2} (F_{q}) \times (F_{q}^{\times})^{n - 2}$ attached to $s$ , the principal series of the restriction of $η$ is $[(a \circ \det) \oplus (St \otimes a \circ \det)] ⊠ ξ$ with $ξ$ a one-dimensional character of $(F_{q}^{\times})^{n - 2}$ ; in particular its constituents have degrees $1$ and $q$ , and the constituent of degree $1$ is isomorphic to $ρ$ restricted to $M$ (The equal-coordinate rank-one principal series of GL_2, Compositions, partial flags, and standard parabolics, Block Levi decomposition of standard parabolics).

[F5]

On the $M_{2} = {GL}_{2} (F_{q})$ factor of the rank-one Levi, the idempotent models of $I (1)$ and $I (a, a)$ are $C [M_{2}] e_{B_{2}}$ and $C [M_{2}] e_{ψ}$ , respectively (Principal series endomorphisms as the chi-idempotent corner). The spherical operator $R_{T_{s}}$ has eigenvalues $q$ on the trivial constituent and $- 1$ on $St$ (The equal-coordinate rank-one principal series of GL_2).

Proof

technique · direct

1.1F1F2F3algebra

Fix $s$ in block $r$ . Apply the algebra automorphism $m_{r}$ of [F2] to the block identity $T_{s}^{(r) 2} = (q - 1) T_{s}^{(r)} + q e_{B_{r}}$ of [F1]: since $m_{r}$ is multiplicative and $m_{r} (T_{s}^{(r)}) = ψ_{r} (\dot{s})^{- 1} Θ_{s}^{(r)}$ , one gets $ψ_{r} (\dot{s})^{- 2} Θ_{s}^{(r) 2} = (q - 1) ψ_{r} (\dot{s})^{- 1} Θ_{s}^{(r)} + q e_{r}$ , that is $Θ_{s}^{(r) 2} = (q - 1) ψ_{r} (\dot{s}) Θ_{s}^{(r)} + q e_{r}$ because $ψ_{r} (\dot{s})^{2} = 1$ .

1.2F2F4F5algebra

Write the rank-one Levi as $M_{2} \times D$ with $M_{2} = {GL}_{2} (F_{q})$ and $D = (F_{q}^{\times})^{n - 2}$ carrying the character $ξ$ of [F4]. Covariant functions on this product satisfy $f (g, d) = ξ (d)^{- 1} f (g, 1)$ , identifying its principal series with $I_{M_{2}} (a, a) ⊠ ξ$ . Put $ψ = a \circ \det$ on $M_{2}$ . The map $m (g) = ψ (g)^{- 1} g$ of [F2] sends $C [M_{2}] e_{B_{2}}$ bijectively to $C [M_{2}] e_{ψ}$ and satisfies $m (h v) = ψ (h)^{- 1} h m (v)$ . It therefore identifies $I_{M_{2}} (1) \otimes ψ$ with $I_{M_{2}} (a, a)$ and intertwines $R_{T_{s}}$ with $R_{m (T_{s})}$ , since $m (v T_{s}) = m (v) m (T_{s})$ . Here $m (T_{s}) = a (- 1)^{- 1} Θ_{s}$ , the normalized operator. By [F5] its eigenvalues are $q$ and $- 1$ on the respective degree- $1$ and degree- $q$ constituents; tensoring with $ξ$ preserves these scalars. The raw intertwiner thus acts by $a (- 1) q$ and $- a (- 1)$ on those constituents.

2.1F3step 1.1algebra

On the module $e_{r} C [{GL}_{n_{r}}] e_{r}$ the idempotent $e_{r}$ acts as the identity, so the operator $R_{Θ_{s}^{(r)}}$ on this module satisfies $R_{Θ_{s}^{(r)}}^{2} = (q - 1) ψ_{r} (\dot{s}) R_{Θ_{s}^{(r)}} + q i d$ ; in the ambient corner $e_{η}^{G} C [G] e_{η}^{G}$ the element $Θ_{s}^{(r)}$ is corrected by the idempotent $e_{U_{P}}$ , whose right multiplication acts as the identity on $C [G] e_{η}^{G}$ , so the raw operator $B_{s}$ on $I (η)$ satisfies $B_{s}^{2} = (q - 1) a_{r} (- 1) B_{s} + q i d$ with $a_{r} (- 1) = ψ_{r} (\dot{s})$ by [F3].

3.1step 2.1algebra

Put $T_{s} = ρ (\dot{s})^{- 1} B_{s} = a_{r} (- 1)^{- 1} B_{s}$ , which is the normalization of Length-additive products of the standard intertwiners; since $a_{r} (- 1)^{2} = 1$ , multiplying the relation of step 2.1 by $a_{r} (- 1)^{- 2}$ gives $T_{s}^{2} = (q - 1) T_{s} + q i d$ . Hence the polynomial $(X - q) (X + 1)$ annihilates $T_{s}$ , so every eigenvalue of $T_{s}$ on any finite-dimensional constituent of $I (η)$ lies in ${q, - 1}$ , and every eigenvalue of the raw operator $B_{s} = a_{r} (- 1) T_{s}$ lies in ${a_{r} (- 1) q, - a_{r} (- 1)}$ .

4.1F4step 3.1givenalgebra

For the normalization, $ρ (\dot{w}) = \prod_{r} a_{r} (\det P_{w_{r}}) = \prod_{r} a_{r} (- 1)^{ℓ (w_{r})}$ for $w \in W_{η}$ , so the factor $λ_{w} = ρ (\dot{w})^{- 1}$ of Length-additive products of the standard intertwiners is exactly $\prod_{r} a_{r} (- 1)^{- ℓ (w_{r})}$ ; for an arbitrary $χ$ the transported generators of that lemma inherit the relation $T_{s}^{2} = (q - 1) T_{s} + q i d$ through the sorting isomorphism. For $q = 3$ and the nontrivial character $a$ of $F_{3}^{\times}$ one has $a (- 1) = a (2) = - 1$ , so the raw eigenvalue on $a \circ \det$ is $a (- 1) q = - 3$ , which is neither $3$ nor $- 1$ : the naive generator fails the parameter- $q$ relation.

5.1step 1.1step 1.2step 2.1step 3.1step 4.1∎

Steps 1.1, 2.1 and 3.1 compute the rank-one quadratic relation with parameter exactly $q$ after normalization; step 1.2 identifies the two constituent degrees and eigenvalues, and step 4.1 records the normalizing cocharacter $λ_{w}$ , the transport to arbitrary $χ$ and the explicit $q = 3$ failure of the naive normalization. All groups, idempotents and eigenvalues are explicit and finite, and no choice principle is used.

Depends on

Used by

The endomorphism algebra of a general finite principal series Theorem

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Sources

Olivier Dudas and Jean Michel, Lectures on Finite Reductive Groups and Their Representations - Lemma 11.10, its proof, and the discussion of (P1)-(P3) preceding Theorem 11.11, printed pp. 49-50 (standard reference, not scraped)
Masao Oi, Representation Theory of Finite Groups of Lie Type - Proposition 2.8(2) and its proof (the equal-character rank-one decomposition), printed pp. 12-13 (standard reference, not scraped)
Jay Taylor, Finite Reductive Groups - Exercise 5.11 ($q_s=q$ for the standard Frobenius and $GL_n$), printed p. 44 (standard reference, not scraped)