LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6.1-sol)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Principal series endomorphisms as the chi-idempotent corner

Statement

Let $n \geq 1$ , let $q$ be a prime power, put $G = {GL}_{n} (F_{q})$ with Borel $B = T ⋉ U$ , let $χ \in \hat{T}$ with inflation $\tilde{χ}$ to $B$ , and let $e_{χ} : = \frac{1}{∣ B ∣} \sum_{b \in B} \tilde{χ} (b)^{- 1} b \in C [B] \subseteq C [G],$ the idempotent of the one-dimensional representation $\tilde{χ}$ of $B$ , so that $b e_{χ} = e_{χ} b = \tilde{χ} (b) e_{χ}$ for $b \in B$ . Then:

the map $C [G] e_{χ} \to I (χ)$ , $g e_{χ} \mapsto f_{g}$ , where $f_{g} (g b) = \tilde{χ} (b)^{- 1}$ and $f_{g} = 0$ outside $g B$ , is an isomorphism of left $C [G]$ -modules;
right multiplication defines an algebra isomorphism $e_{χ} C [G] e_{χ} \tilde{\to} {End}_{C [G]} (C [G] e_{χ})^{o p} ≅ {End}_{G} (I (χ))^{o p};$
writing $\dot{w}$ for the permutation matrix of $w \in S_{n}$ , the elements $e_{χ} \dot{w} e_{χ}$ , $w \in S_{n}$ , span $e_{χ} C [G] e_{χ}$ ; one has $e_{χ} \dot{w} e_{χ} = 0$ whenever $w \notin W_{χ}$ , and the elements $e_{χ} \dot{w} e_{χ}$ with $w \in W_{χ}$ form a $C$ -basis of $e_{χ} C [G] e_{χ}$ . Hence $\dim_{C} {End}_{G} (I (χ)) = ∣ W_{χ} ∣$ , with basis indexed by the Weyl stabiliser, in accordance with The Weyl stabiliser controls the principal series endomorphisms.

No choice principle is used beyond the finite selection of coset representatives used to exhibit a basis.

Facts & Assumptions

Given: $G = {GL}_{n} (F_{q})$ with Borel $B = T ⋉ U$ , a character $χ \in \hat{T}$ with inflation $\tilde{χ}$ , the idempotent $e_{χ}$ , the corner $e_{χ} C [G] e_{χ}$ and the module $I (χ)$ .

[F1]

The group algebra $C [G]$ has basis the group elements and unit $1$ ; for $b \in B$ one has $b e_{χ} = e_{χ} b = \tilde{χ} (b) e_{χ}$ , and $e_{χ}^{2} = e_{χ}$ (The group ring $R [G]$ is a unital $R$ -algebra with basis $G$ , and each $g \in G$ is a unit of $R [G]$ ).

[F2]

The induced module $I (χ) = {Ind}_{B}^{G} (\tilde{χ})$ is the $C$ -vector space of covariant functions with the left action $(h \cdot f) (x) = f (h^{- 1} x)$ (The induced $R$ -linear $G$ -module ${Ind}_{H}^{G} W$ as $H$ -covariant functions on $G$ , The principal series module for finite GL_n). If $T = {t_{1}, \dots, t_{n}}$ meets each left coset $g B$ in exactly one point, then evaluation at $T$ is an isomorphism ${Ind}_{B}^{G} (\tilde{χ}) \to ⨁_{t \in T} C$ , so the functions $f_{t}$ with $f_{t} (t b) = \tilde{χ} (b)^{- 1}$ and $f_{t} = 0$ outside $t B$ form a $C$ -basis of $I (χ)$ (A left transversal identifies ${Ind}_{H}^{G} W$ with a direct sum of $[G : H]$ copies of $W$ ).

[F3]

Endomorphisms of a module form a ring under pointwise addition and composition (Module endomorphisms form a ring under pointwise addition and composition).

[F4]

The double cosets $B \dot{w} B$ , $w \in S_{n}$ , partition $G$ (Bruhat decomposition of GL_n over a finite field), and the Weyl stabiliser $W_{χ} = {w : w \cdot χ = χ}$ satisfies $\dim_{C} {End}_{G} (I (χ)) = ∣ W_{χ} ∣$ (Diagonal torus characters and the Weyl action, The Weyl stabiliser controls the principal series endomorphisms).

Proof

technique · direct

1.1F1algebra

For $b \in B$ , reindexing $c = b b'$ in the sum defining $b e_{χ}$ gives coefficient $\tilde{χ} (b^{- 1} c)^{- 1} = \tilde{χ} (b) \tilde{χ} (c)^{- 1}$ ; reindexing $c = b' b$ gives the same coefficient for $e_{χ} b$ . Thus $b e_{χ} = e_{χ} b = \tilde{χ} (b) e_{χ}$ , and $e_{χ}^{2} = ∣ B ∣^{- 1} \sum_{b} \tilde{χ} (b)^{- 1} \tilde{χ} (b) e_{χ} = e_{χ}$ .

2.1F2step 1.1construct

The assignment $Φ (g e_{χ}) : = f_{g}$ with $f_{g} (g b) = \tilde{χ} (b)^{- 1}$ and $f_{g} = 0$ off $g B$ is well defined on $C [G] e_{χ}$ : by step 1.1, $g b e_{χ} = \tilde{χ} (b) g e_{χ}$ for $b \in B$ , while $f_{g b} = \tilde{χ} (b) f_{g}$ : at $g b b'$ the left side equals $\tilde{χ} (b')^{- 1}$ , and the right side equals $\tilde{χ} (b) \tilde{χ} (b b')^{- 1}$ , so both sides scale in the same way along right $B$ -orbits. It is $C [G]$ -linear because $f_{h g} = h \cdot f_{g}$ for $g, h \in G$ by the left action formula of [F2], and it is bijective: for a finite set $T$ of left coset representatives the elements $t e_{χ}$ , $t \in T$ , form a $C$ -basis of $C [G] e_{χ}$ (every element is a combination of the $g e_{χ}$ , and $g e_{χ}$ is a nonzero scalar multiple of the chosen representative vector for $g B$ by step 1.1, and the representative vectors have disjoint coset supports), while the $f_{t}$ , $t \in T$ , form a $C$ -basis of $I (χ)$ by [F2]; as $Φ (t e_{χ}) = f_{t}$ , it maps one basis to the other. This proves (1).

2.2F4step 1.1algebra

The double cosets $B \dot{w} B$ partition $G$ by [F4], so $e_{χ} C [G] e_{χ}$ is spanned by the elements $e_{χ} g e_{χ}$ with $g \in G$ ; for $b, b^{'} \in B$ one has $e_{χ} (b x b^{'}) e_{χ} = \tilde{χ} (b) \tilde{χ} (b^{'}) e_{χ} x e_{χ}$ by step 1.1, so each cell contributes the single vector $e_{χ} \dot{w} e_{χ}$ up to a nonzero scalar, and the $e_{χ} \dot{w} e_{χ}$ , $w \in S_{n}$ , span the corner. If $w \notin W_{χ}$ then there is $t \in T$ with $χ (t) \neq χ ({\dot{w}}^{- 1} t \dot{w})$ ; from ${\dot{w}}^{- 1} t \dot{w} \in B$ and step 1.1 one has $e_{χ} t \dot{w} e_{χ} = \tilde{χ} (t) e_{χ} \dot{w} e_{χ}$ and also $e_{χ} t \dot{w} e_{χ} = e_{χ} \dot{w} ({\dot{w}}^{- 1} t \dot{w}) e_{χ} = \tilde{χ} ({\dot{w}}^{- 1} t \dot{w}) e_{χ} \dot{w} e_{χ}$ , so the differing scalars force $e_{χ} \dot{w} e_{χ} = 0$ .

3.1F3step 1.1step 2.1algebra

Let $f : C [G] e_{χ} \to C [G] e_{χ}$ be $C [G]$ -linear and put $a : = f (e_{χ})$ . Then $f (g e_{χ}) = g a$ , and $e_{χ} a = a = f (e_{χ}) = f (e_{χ}^{2}) = e_{χ} a$ , while $a = f (e_{χ}) \in C [G] e_{χ}$ gives $a e_{χ} = a$ ; hence $a \in e_{χ} C [G] e_{χ}$ . Conversely for $a \in e_{χ} C [G] e_{χ}$ the right multiplication $ρ_{a} (y e_{χ}) : = y e_{χ} a$ maps $C [G] e_{χ}$ to itself and commutes with left multiplication by $C [G]$ , and $ρ_{a} ρ_{b} = ρ_{b a}$ . So $f \mapsto f (e_{χ})$ is a $C$ -linear bijection from ${End}_{C [G]} (C [G] e_{χ})$ onto $e_{χ} C [G] e_{χ}$ whose inverse reverses composition, i.e. an algebra isomorphism onto the opposite corner; transporting along the isomorphism of step 2.1 identifies ${End}_{C [G]} (C [G] e_{χ})$ with ${End}_{G} (I (χ))$ . This proves (2) up to the transport.

4.1F4step 2.2step 3.1algebra

By step 3.1 the corner has dimension $\dim {End}_{G} I (χ) = ∣ W_{χ} ∣$ from [F4]. Step 2.2 spans it by the $∣ W_{χ} ∣$ vectors $e_{χ} \dot{w} e_{χ}$ with $w \in W_{χ}$ . A spanning family of exactly the dimension of a finite-dimensional space is a basis, so all these vectors are nonzero and linearly independent. This proves (3) without assuming that permutation matrices normalize the Borel subgroup.

5.1F4step 2.1step 3.1step 2.2step 4.1∎

Clause (1) is step 2.1, clause (2) is step 3.1, and clause (3) is steps 2.2 and 4.1; the identification of dimension with $∣ W_{χ} ∣$ agrees with the independent computation of [F4]. The only selection made is a finite set of left coset representatives in step 2.1, which exists by finite choice for the finitely many cosets, and the double-coset representatives are the explicit permutation matrices; no infinite choice is used.

Depends on

Used by

Dependency tree · two levels

38 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Ivan Losev, Lecture 8: Representations of GL_n(F_q) - Section 2.1, Lemma 2.1 and its proof (the idempotent corner e C[B\G] e), PDF pp. 3-4 (standard reference, not scraped)
Olivier Dudas and Jean Michel, Lectures on Finite Reductive Groups and Their Representations - Section 11.1 (the corner H = e_{B^F} Lambda G^F e_{B^F}), printed p. 46 (standard reference, not scraped)