The character of the principal series of GL₂(𝔽_q) on the four families of conjugacy classes #
The conjugacy classes of GL₂(𝔽_q) fall into four families
(TauCeti/LinearAlgebra/Matrix/GeneralLinearGroup/ConjugacyClasses.lean), with representatives
the central scalar diag(a, a), the split semisimple diag(a, b) with a ≠ b, the
non-semisimple Jordan block !![a, b; 0, a] with b ≠ 0, and the elliptic image of an
element of E ∖ F under the non-split torus of a quadratic extension E/F. This file evaluates
the character of the principal series Ind_B^{GL₂}(α ⊗ β) at each of the four, giving the row
χ_{α,β} = (q + 1) α(a) β(a), α(a) β(b) + α(b) β(a), α(a) β(a), 0
of the character table of GL₂(𝔽_q). At α = β = 1 these are the fixed-point counts on the
projective line, which is the boundary case
TauCeti/RepresentationTheory/CharacterTable/GL2/CharacterValues.lean reads off.
The computation is the induced-character formula in the form
Subgroup.indClassFun_eq_sum_of_smul_eq_self_mem: the character of an induction at g is the sum
of the inducing character over the cosets of B that g fixes, evaluated at the conjugate of g
into B that each such coset exhibits. Which cosets are fixed is already known — a scalar fixes
all q + 1 of them, a split semisimple element exactly 2, a Jordan block exactly 1, and an
elliptic element none
(TauCeti/LinearAlgebra/Matrix/GeneralLinearGroup/ProjectiveLine.lean) — and those counts are
enough to identify the fixed cosets, because in each case a list of that many fixed cosets is at
hand: the trivial coset B and, for a split semisimple element, the coset of the Weyl element
w = !![0, 1; 1, 0]. Conjugating diag(a, b) by w swaps the two entries, which is where the
second summand α(b) β(a) comes from.
Only the central formula contains q, as the index q + 1 of the Borel subgroup, and it is the
one whose fixed-coset count TauCeti.GL2Borel.natCard_fixedCosets_scalar is about a finite field.
The other three counts hold over any field; all four proofs still need F finite, because the
coset sum they run on is the one of a finite-index subgroup.
Main results #
TauCeti.character_GL2PrincipalSeries_scalar,TauCeti.character_GL2PrincipalSeries_diagGL,TauCeti.character_GL2PrincipalSeries_jordanGLandTauCeti.character_GL2PrincipalSeries_gl2NonSplitTorusHom: the four values above.
References #
- W. Fulton and J. Harris, Representation Theory: A First Course, GTM 129, §5.2.
- C. Bonnafé, Representations of
SL₂(𝔽_q)(2011), Chapter 5.
The summands of the induced-character formula #
The four values #
The principal series has character (q + 1) α(a) β(a) at a central element. A scalar
matrix is central, so every one of the q + 1 cosets of the Borel subgroup is fixed, and each
contributes the value of α ⊗ β at the scalar itself.
The principal series has character α(a) β(b) + α(b) β(a) at a split semisimple element.
A diagonal matrix with distinct entries fixes exactly two cosets of the Borel subgroup, the
trivial one and the one of the Weyl element; conjugating by the Weyl element swaps the two
diagonal entries, so the two cosets contribute α(a) β(b) and α(b) β(a).
The principal series has character α(a) β(a) at a non-semisimple element. A Jordan block
fixes exactly one coset of the Borel subgroup, the trivial one, and it is upper triangular with
both diagonal entries equal to a.
The principal series vanishes at an elliptic element. An element of the non-split torus
coming from E ∖ F has no eigenline over F, so it fixes no coset of the Borel subgroup and the
induced-character sum is empty.