The non-split torus of GL₂ #
Let E/F be a field extension of degree 2. Choosing an F-basis of E presents multiplication
by an element of E as a 2 × 2 matrix over F, and multiplication by a nonzero element as an
element of GL (Fin 2) F. The image of Eˣ is the non-split torus
TauCeti.GL2NonSplitTorus F E, an abelian subgroup of GL₂(F) of order q² - 1 when F has
q elements. It is the torus the cuspidal (discrete series) representations of GL₂(𝔽_q) are
parametrized by: a character of Eˣ in general position determines one of them through the
Deligne–Lusztig construction, which is not ordinary induction (the induced representation
Ind_{Eˣ}^{GL₂(𝔽_q)} has dimension q(q - 1), while a cuspidal representation has dimension
q - 1). This is the elliptic counterpart of the split torus of diagonal matrices, whose
characters give the principal series by ordinary induction from the Borel subgroup.
The word torus is the algebraic-group one only when E/F is separable, which over a finite
field — the setting of the roadmap target — it always is. Degree 2 alone also admits a purely
inseparable E/F (characteristic 2 only), where the algebra E ⊗[F] F̄ is the nonreduced
F̄[X]/(X²) rather than F̄ × F̄. The image of Eˣ is then not a torus: after base change to
F̄ it is the unit group of F̄[X]/(X²), which is Gₘ × Gₐ — still smooth and reduced as a
group, but with a nontrivial unipotent part — and its non-scalar elements are not semisimple.
Everything stated below, non-splitness included, is true in that case too: no proof here uses
separability, so no statement assumes it.
"Non-split" is the assertion that, away from the scalars, the torus is not conjugate into the Borel
subgroup: TauCeti.GL2NonSplitTorus.conj_notMem_gl2Borel says that if x : Eˣ does not lie in F
then no conjugate of its matrix is upper triangular. Equivalently the matrix has no eigenvalue in
F — over a finite field its eigenvalues are a conjugate pair in E ∖ F — which is what makes
these the elliptic conjugacy classes of GL₂(𝔽_q). The proof is that the determinant of
x - a is the norm N_{E/F}(x - a), which is nonzero because x - a is.
The construction depends on the chosen basis TauCeti.nonSplitTorusBasis; a different choice
conjugates the subgroup, so the statements that are not conjugation-invariant are stated for this
choice, following the convention of
TauCetiRoadmap/RepresentationTheory/CharacterTheory/README.md.
Main definitions #
TauCeti.nonSplitTorusBasis: a chosenF-basis ofE, indexed byFin 2.TauCeti.GL2NonSplitTorusHom: the embeddingEˣ ↪ GL (Fin 2) Fby left multiplication.TauCeti.GL2NonSplitTorus: its range, the non-split torus.TauCeti.GL2NonSplitTorus.unitsEquiv: the resulting multiplicative equivalenceEˣ ≃*the torus.
Main results #
TauCeti.GL2NonSplitTorus.natCard_eq: the torus hasq² - 1elements over a finite field withqelements, andTauCeti.GL2NonSplitTorus.index_eq: its index is thenq (q - 1).TauCeti.GL2NonSplitTorus.notMem_range_scalar_gl2NonSplitTorusHom: an element of the torus not coming fromFis not a scalar matrix, hence is regular.TauCeti.GL2NonSplitTorus.conj_notMem_of_det_sub_algebraMap_eq_zero: a non-scalar element with an eigenvalue inFhas no conjugate in the torus. In particular,TauCeti.GL2NonSplitTorus.conj_notMem_gl2Borelsays an element of the torus not coming fromFhas no conjugate in the Borel subgroup, andTauCeti.GL2NonSplitTorus.exists_forall_conj_notMem_gl2Borel: such an element exists, so the torus is not conjugate into the Borel subgroup. What these contradict isTauCeti.GL2Borel.exists_det_sub_algebraMap_eq_zero, that a matrix with an upper-triangular conjugate has an eigenvalue in the base ring.
References #
- Character theory roadmap, Layer 9.
- W. Fulton and J. Harris, Representation Theory: A First Course (1991), Lecture 5.2.
- C. Bonnafé, Representations of
SL₂(𝔽_q)(2011), Chapter 1.
A chosen F-basis of a degree-2 extension E/F, indexed by Fin 2. The non-split torus is
the image of Eˣ under the matrix representation in this basis; another choice of basis conjugates
it.
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The embedding of the non-split torus: a nonzero element of a degree-2 extension E/F
acts on E by multiplication, hence, in the basis TauCeti.nonSplitTorusBasis, as an element of
GL (Fin 2) F.
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The non-split (elliptic) torus of GL₂(F) attached to a degree-2 extension E/F: the
image of Eˣ under multiplication on E, read in the basis TauCeti.nonSplitTorusBasis. It is a
torus in the algebraic-group sense when E/F is separable, in particular whenever F is finite;
for a purely inseparable E/F it is the same subgroup, still non-split in the sense proved below,
but not an algebraic torus (see the module docstring).
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Membership in the non-split torus: a matrix lies in it exactly when it is left multiplication
by a unit of E.
The matrix underlying GL2NonSplitTorusHom F E x is multiplication by x in the basis
TauCeti.nonSplitTorusBasis.
Distinct elements of Eˣ give distinct matrices.
The non-split torus is a copy of Eˣ: the embedding TauCeti.GL2NonSplitTorusHom is
injective, so it corestricts to a multiplicative equivalence from Eˣ onto the torus. This is what
transports a character of Eˣ to a character of the torus.
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TauCeti.GL2NonSplitTorus.unitsEquiv is TauCeti.GL2NonSplitTorusHom on the nose.
The torus is abelian: it is the image of the commutative group Eˣ.
The determinant of a torus element is the norm of the field element it comes from.
The trace of a torus element is the trace of the field element it comes from.
The determinant of a non-split-torus element is its field norm, as an equality of units.
A unit of F is sent to the corresponding scalar matrix.
The scalar matrices lie in the non-split torus: it contains the centre of GL₂(F).
A scalar matrix, read back through TauCeti.GL2NonSplitTorus.unitsEquiv, is the unit of F
it came from, pushed into E.
The order of the non-split torus: it has one element for each nonzero element of E, so
over a field with q elements it has q² - 1 of them. (Over an infinite F both sides are 0,
the Nat.card of an infinite type.)
The index of the non-split torus: over a field with q elements the torus has q² - 1
elements inside a group of order (q² - 1) q (q - 1), so its index is q (q - 1). It is the
number of summands in a class function induced from the torus, and hence the dimension of a
representation induced from a character of Eˣ.
An element of the non-split torus not coming from F is not a scalar matrix: multiplication by
x on E is multiplication by an element of F exactly when x lies in F. It is therefore
regular, which is what makes its centralizer computable.
The key computation behind non-splitness: for x : E outside F, the matrix of multiplication
by x has no eigenvalue a : F.
A non-scalar element with an eigenvalue in F has no conjugate in the non-split torus.
An element of the torus is either scalar or has no eigenvalue in F
(TauCeti.GL2NonSplitTorus.det_sub_algebraMap_ne_zero), and both conditions are invariant under
conjugation.
The torus is non-split: if x : Eˣ does not come from F, then no conjugate of the
corresponding matrix is upper triangular. Equivalently, that matrix has no eigenvalue in F, which
over a finite field is what makes its conjugacy class elliptic.
The non-split torus is not conjugate into the Borel subgroup: it contains an element no conjugate of which is upper triangular. This is exactly what distinguishes it from the split torus of diagonal matrices, which lies in the Borel subgroup outright, and it is why the cuspidal representations attached to it are absent from every principal series.