Paper XI: Finite field cuspidal projectors

4 Finite field cuspidal projectors

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Let

\[ G_n={\operatorname{GL}}_n({\mathbb F}_q), \qquad T_n\cong{\mathbb F}_{q^n}^{\times} \]

with \(T_n\) an elliptic maximal torus.

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Standard fact 4.1: Regular elliptic torus characters

A regular character \(\theta:T_n\to{\mathbb C}^{\times}\) determines an irreducible cuspidal representation \(\kappa_\theta\) of \(G_n\). Frobenius conjugate characters determine isomorphic representations, and every irreducible cuspidal representation of \(G_n\) occurs in this way.

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This is the finite field input from Deligne and Lusztig theory [13]. We use standard finite group character theory for the central idempotents [28], [21]. Once \(\kappa_\theta\) is known, its block is selected by a central idempotent.

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Proposition 4.2: Primitive cuspidal idempotent

Let \(\chi_\theta\) be the irreducible character of \(\kappa_\theta\), and let \(d_\theta=\chi_\theta(1)\). Then

\[ e_\theta = \frac{d_\theta}{{\left\lvert G_n\right\rvert}} \sum_{g\in G_n} \chi_\theta(g^{-1})g \]

is the primitive central idempotent selecting the \(\kappa_\theta\) block in \({\mathbb C}[G_n]\).

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Proof of Proposition 4.2

On an irreducible representation of character \(\psi\), the central element acts by a scalar. Taking traces shows that the scalar is

\[ \frac{d_\theta}{{\left\lvert G_n\right\rvert}\psi(1)} \sum_{g\in G_n} \chi_\theta(g^{-1})\psi(g). \]

Character orthogonality makes this scalar one for \(\psi=\chi_\theta\) and zero otherwise. The element is therefore central, idempotent, and primitive.

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Proposition 4.3: Parabolic annihilation

Let \(P=LU\) be a proper parabolic subgroup of \(G_n\), and put

\[ e_U=\frac1{{\left\lvert U\right\rvert}}\sum_{u\in U}u. \]

If \(\kappa_\theta\) is cuspidal, then

\[ \boxed{ e_Ue_\theta=e_\theta e_U=0. } \]

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Proof of Proposition 4.3

The operator \(e_U\) projects onto the subspace of \(U\) fixed vectors. A cuspidal representation has no nonzero fixed vector under the unipotent radical of a proper parabolic. Thus \(e_U\) acts by zero on the \(\kappa_\theta\) block. Since \(e_\theta\) is central, both products vanish.

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A regular torus character determines a cuspidal block and its primitive central projector.

4.1 Depth zero reconstruction and cycle orientation

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The depth zero construction inflates a finite cuspidal representation to \({\operatorname{GL}}_n({\mathcal O}_K)\), extends it to the standard compact modulo center subgroup, and then compactly induces it to \({\operatorname{GL}}_n(K)\). This is classical type theory [11], [8].

The induced tame Weil atom has an elementary cycle invariant. In an orbit basis \(e_0,\ldots,e_{n-1}\), write

\[ \tau(\Phi)e_j=e_{j+1} \]

for \(j<n-1\), and

\[ \tau(\Phi)e_{n-1}=ae_0. \]

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Proposition 4.4: Frobenius cycle polynomial

The Frobenius transfer matrix has characteristic polynomial

\[ X^n-a \]

and determinant

\[ \boxed{ \det\tau(\Phi)=(-1)^{n-1}a. } \]

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Proof of Proposition 4.4

The matrix is the companion matrix of \(X^n-a\). Its only nonzero determinant term is the closed \(n\) cycle. That cycle has sign \((-1)^{n-1}\), and the closing edge contributes \(a\).

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Corollary 4.5: Unramified orientation character

The sign \((-1)^{n-1}\) is the orientation of the closed Frobenius packet cycle. It is unchanged by rescaling the individual orbit basis vectors.

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Proof of Corollary 4.5

A basis rescaling conjugates the transfer matrix by a diagonal matrix, so the determinant and characteristic polynomial remain unchanged.

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