The linear Harish-Chandra projection and the Weyl invariants #
Let L be a finite-dimensional Lie algebra with non-degenerate Killing form over a field K of
characteristic zero, H a splitting Cartan subalgebra and b a base of its root system, with Weyl
vector ρ. The Harish-Chandra projection TauCeti.hcProjection b : Z(U(L)) →ₐ[K] S(H) reads the
central character χ_λ as the polynomial function λ ↦ χ_λ(z) on the weights, and these functions
are invariant under the dot action w · λ = w(λ + ρ) - ρ of the Weyl group
(TauCeti.hcProjection_mem_dotInvariants). This file passes to the linear normalization, in
which the invariance is under the ordinary linear action of the Weyl group on the weights.
The passage is translation by -ρ, SymmetricAlgebra.translation, the automorphism of
S(H) with (translation μ p)(λ) = p(λ + μ). Since w · (λ - ρ) = w λ - ρ, translation by -ρ
carries the dot-invariants S(H)^{W·} (TauCeti.dotInvariants) onto the linear invariants
S(H)^W (TauCeti.linearInvariants), the polynomial functions p with p(w λ) = p(λ); this is
TauCeti.dotInvariantsEquivLinearInvariants. Like the dot-invariants, S(H)^W is defined by
evaluation at the weights; since the action is now linear, it is also the invariant subalgebra of
the action of the Weyl group on H, the coroot space
(TauCeti.linearInvariants_eq_symmetricAlgebraInvariants). The linear Harish-Chandra
projection
TauCeti.hcProjectionLinear b = translation (-ρ) ∘ hcProjection b : Z(U(L)) →ₐ[K] S(H)
therefore lands in S(H)^W, and the central character is its value at the ρ-shifted weight:
χ_λ(z) = (hcProjection b z)(λ) = (hcProjectionLinear b z)(λ + ρ).
These two equations fix the sign of the shift. The linear normalization is the one in which the
Harish-Chandra isomorphism Z(U(L)) ≃ S(H)^W is compared with the Chevalley restriction theorem
for S(L)^L: there the Weyl group acts linearly, so this is the form in which the two are matched.
Over an algebraically closed field the projection is onto the Weyl invariants
(TauCeti.range_hcProjectionLinear).
Main definitions #
TauCeti.linearInvariants H: the Weyl invariantsS(H)^W, the elementspofS(H)withp(w λ) = p(λ)for every Weyl group elementwand weightλ.TauCeti.dotInvariantsEquivLinearInvariants b: translation by-ρ, as an algebra isomorphism from the dot-invariants onto the Weyl invariants.TauCeti.hcProjectionLinear b: the linear Harish-Chandra projectionZ(U(L)) →ₐ[K] S(H), the Harish-Chandra projection followed by translation by-ρ.
Main results #
TauCeti.linearInvariants_eq_symmetricAlgebraInvariants: the Weyl invariants are the invariants of the coroot representationRootPairing.weylGroupCorootReponH.TauCeti.isHomogeneous_linearInvariants: every homogeneous component of a Weyl invariant is a Weyl invariant.TauCeti.translation_neg_weylVector_mem_linearInvariants_iff: translation by-ρtakes an element ofS(H)to a Weyl invariant exactly when the element is dot-invariant.TauCeti.lift_hcProjectionLinearandTauCeti.lift_add_weylVector_hcProjectionLinear: the value of the linear projection atλis the central character ofλ - ρ; equivalently,χ_λ(z)is its value atλ + ρ.TauCeti.hcProjectionLinear_mem_linearInvariants: the linear projection takes values in the Weyl invariants.
References #
- J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, GTM 9, §23.3, where
the Harish-Chandra homomorphism is the projection followed by the translation by
-ρ. - J. E. Humphreys, Representations of Semisimple Lie Algebras in the BGG Category
O, Chapter 1, on Harish-Chandra's theorem.
The Weyl invariants #
The Weyl invariants S(H)^W: the elements p of the symmetric algebra S(H), read as
polynomial functions on the weights H*, with p(w lam) = p(lam) for every Weyl group element w
and every weight lam, where w lam is the linear action.
Instances For
Membership in the Weyl invariants: p(w lam) = p(lam) for every Weyl group element w and
every weight lam.
Comparison with the invariants of the coroot representation #
The Weyl invariants are the invariants of the coroot representation: S(H)^W, defined by
evaluation at the weights, is the invariant subalgebra of S(H) under the representation
RootPairing.weylGroupCorootRep of the Weyl group on H. In particular it inherits the
invariant-theoretic API of Representation.symmetricAlgebraInvariants, such as its homogeneity.
The Weyl invariants are homogeneous: every homogeneous component of a Weyl invariant is a
Weyl invariant, since the Weyl group acts on S(H) by graded automorphisms.
From dot-invariants to Weyl invariants #
Translation by -ρ turns dot-invariance into Weyl invariance: translation (-ρ) p is a
Weyl invariant exactly when p is dot-invariant. The reason is the identity
w · (lam - ρ) = w lam - ρ.
Translation by -ρ carries the dot-invariants onto the Weyl invariants.
The dot-invariants are isomorphic to the Weyl invariants, by translation by -ρ:
p ↦ (lam ↦ p(lam - ρ)).
Instances For
The isomorphism from the dot-invariants to the Weyl invariants is translation by -ρ.
The inverse isomorphism, from the Weyl invariants to the dot-invariants, is translation by
ρ.
The linear Harish-Chandra projection #
The linear Harish-Chandra projection Z(U(L)) →ₐ[K] S(H): the Harish-Chandra projection
TauCeti.hcProjection b followed by translation by -ρ, so that its value at a central element
z is the polynomial function lam ↦ χ_{lam - ρ}(z) (TauCeti.lift_hcProjectionLinear). It
takes values in the Weyl invariants (TauCeti.hcProjectionLinear_mem_linearInvariants).
Instances For
The linear Harish-Chandra projection is the Harish-Chandra projection translated by -ρ.
The value of the linear Harish-Chandra projection at a weight lam is the central
character of lam - ρ.
The central character through the linear Harish-Chandra projection: χ_lam(z) is the
value of hcProjectionLinear b z at lam + ρ.
The linear Harish-Chandra projection takes values in the Weyl invariants.