November 10, 1997

Actions of “nilpotent tori” on G-varieties

Dmitri I. Panyushev

Introduction

Let G be a connected simply-connected semisimple algebraic group with Lie algebra 𝔤 and let e ∈ 𝔤 be a regular nilpotent element. Denote by G_{e} the centralizer of e in G. It is well known that G_{e} ≃ G_{e}^{o} × Z(G), where Z(G) is the centre of G and G_{e}^{o} is a commutative unipotent group of dimension rk G, the rank of G. R.K. Brylinski used the name “nilpotent torus” for the Lie algebra 𝔤_{e} := Lie G_{e} [5]. We shall apply the name “nilpotent torus” to the groups G_{e}^{o} and G_{e}, too. The aim of this paper is to study the algebras of invariants k[X]^{G_{e}^{o}} and k[X]^{G_{e}}, where X is an affine variety acted upon by G (= G-variety). The motivation (and an explanation of terminology) comes from the article of R.K. Brylinski [4]. She proved that 𝔤_{e} is the limit of Cartan subalgebras in a very precise sense and that, for any irreducible Gmodule V , V^{𝔤_{e}} together with some additional data keeps information on q-analogs of the weight multiplicities in V . Therefore knowledge of the structure of the above algebras of invariants may provide some information on q-analogs for G-modules involved in k[X]. A useful class of G-varieties which deserves a special attention in this regard is that of HV-varieties, see [13, § 5] and sect. 3. In sect. 1, I tried to gather as many results of general nature on invariants of nilpotent tori as possible. Among them the assertion that Spec k[G]^{G_{e}^{o}} has rational singularities. It implies that Spec k[X]^{G_{e}^{o}} has rational singularities whenever X does so. In sect. 2, we describe the null-cone for G_{e}-invariants in terms of 1-parameter subgroups. This was inspired by the work of M. Brion on the null-cone for a maximal unipotent subgroup of G [2, IV]. In our case the answer turns out to be better and it has an interesting connection with the null-cone for a maximal torus. A surprising consequence is that the null-cone relative to G_{e} is never reduced. Actions of nilpotent tori on HV-varieties are considered in sect. 3. We describe all the cases when the algebra of G_{e}-invariants is polynomial and prove some curious results on G_{e}^{o}-invariants for G = A_{p}. In sect. 4, “G_{e}-theory” is used for getting some results on the diagonal G-action on X × 𝔤. First, an estimate of embedding dimension of algebras of the form k[X × 𝔤]^{G} is given. As a by-product, we recover results of F. Levstein [11]. Second, it is proven that the null-cone in X × 𝔤 relative to G is never reduced. Notation. The ground field k is algebraically closed and of characteristic zero. Lie algebras of algebraic groups are denoted by the respective lower-case Gothik letters. Throughout the paper, X is an affine G-variety and k[X] is the algebra of regular functions on X. For a subgroup A ⊂ G, k[X]^{A} is the subalgebra of A-invariant functions. We write X//A for Spec k[X]^{A} whenever k[X]^{A} is finitely generated.

This paper was written during my stay at the MPI (Bonn). I am grateful to the Institute for hospitality and support. The work was partially supported by RFFI Grant No. 95– 01–00783a.

1 Invariants of nilpotent tori on G-varieties

Most properties of G_{e}-invariants come up through a relationship with G-invariants and the nilpotent cone 𝔑 ⊂ 𝔤. Therefore we begin with recalling basic properties of the latter.

1.1 Theorem.

(i) [10] 𝔑 is an irreducible normal complete intersection in 𝔤 and codim_{𝔑}(𝔑 \ Ge) = 2; (ii) [9] 𝔑 has rational singularities. □

It follows that k[𝔑] = k[Ge] = k[G]^{G_{e}}, i.e., k[G]^{G_{e}} is finitely generated and 𝔑 = Spec k[G]^{G_{e}}. Consider the larger algebra k[G]^{G_{e}^{o}}. Being a finite k[G]^{G_{e}} -module, k[G]^{G_{e}^{o}} is finitely generated, too. Obviously, it is integrally closed. Hence 𝔑̃ := Spec k[G]^{G_{e}^{o}} is a normal affine variety. The dense G-orbit in 𝔑̃ is isomorphic to G/G_{e}^{o} and its complement is of codimension 2. Let ẽ be a point in this G-orbit whose stabilizer is G_{e}^{o}.

1.2 Theorem. 𝔑̃ has rational singularities.

Proof. We follow the same way as in [12], where it was proved that the normalization of the closure of any nilpotent orbit has rational singularities. The reader may consult that paper for more details concerning the following exposition. Let K_{Y} and ω_{Y} denote, respectively, the Grauert-Riemenschneider canonical sheaf and the Grothendieck dualizing sheaf on a normal variety Y . By Flenner’s theorem, 𝔑̃ has rational singularities whenever ω𝔑̃ is invertible and K𝔑̃ = ω𝔑̃. Here we have ω𝔑̃ ≃ O𝔑̃ for the same reason as for the closure of any nilpotent orbit (cf. [12]): the dense orbit G · ẽ has a symplectic structure and its complement is of codimension 2. Let τ̃ : Z → 𝔑̃ be a G-equivariant resolution of singularities. In our setting, the equality K𝔑̃ = ω𝔑̃ will follow from the fact that the pull-back of the symplectic 2-form on G · ẽ under τ̃ extends to a regular 2-form on Z. Let B be the unique Borel subgroup such that e ∈ 𝔟 and let 𝔲 be the nilpotent radical of 𝔟. Recall the famous Springer resolution τ : G ∗_{B} 𝔲 = G ∗_{B} \overline{B · e} → 𝔑. We may construct Z such that the following diagram exists and is commutative:

\begin{matrix}Z & ⟶^{τ̃} & 𝔑̃ \\ ↓ & & ↓ \\ G ∗_{B} \overline{B · e} & ⟶^{τ} & 𝔑 \end{matrix} .

Indeed, take a B -equivariant resolution of singularities R → \overline{B · ẽ} ⊂ 𝔑̃ and set Z = G ∗_{B} R. Next, the symplectic 2-form on G · ẽ is the pull-back of the Kirillov form on G · e and the latter extends to a regular 2-form on G ∗_{B} \overline{B · e}, see [12, lemma 2]. ¿From this diagram it then follows that the symplectic form on G · ẽ extends to a regular 2-form on Z. □

Remarks. 1. Making use of Graham’s results [7], one can describe an explicit model of 𝔑̃. Let χ_{1}, …, χ_{r} be the nonzero minuscule dominant weights of G and V_{χ_{i}} (i = 1, …, r) the corresponding irreducible G-modules. Let v_{i} ∈ V_{χ_{i}} be a B -eigenvector. Consider the point ẽ = (e, v_{1}, …, v_{r}) ∈ 𝔤 ⊕ V_{χ_{1}} ⊕ … ⊕ V_{χ_{r}}. Clearly, Gẽ = Bẽ = G_{e}^{o}. I claim that \overline{G · ẽ}, the closure of G-orbit of ẽ in 𝔤 ⊕ V_{χ_{1}} ⊕ … ⊕ V_{χ_{r}} is isomorphic to 𝔑̃. Because 𝔑̃ is normal and the complement of the dense orbit is of codimension 2, the identity map of dense orbits gives rise to a dominant regular map 𝔑̃ → \overline{G · ẽ}. In other words, k[\overline{G · ẽ}] is a subalgebra of k[𝔑̃]. By [7, 1.2,3.4], each minuscule representation occurs in k[𝔑̃] exactly once and k[𝔑̃] is generated by these representations as k[𝔑]-module. By construction, k[\overline{G · ẽ}] contains all the stuff just described. Hence it must be equal to k[𝔑̃].

2. Using this model, it is easy to see that \overline{B · ẽ} is isomorphic to the product of \overline{T · ẽ} and an affine space (here T is a maximal torus of B). The closure \overline{T · ẽ} is not normal whenever 𝔑̃ ≠ 𝔑, but its normalization has rational singularities. Therefore, it is enough to take the normalization of \overline{B · ẽ} in place of R in the previous proof.

1.3 Proposition. For any affine G-variety X, we have:

(i) k[X]^{G_{e}} ≃ k[X × 𝔑]^{G} and k[X]^{G_{e}^{o}} ≃ k[X × 𝔑̃]^{G}, where the isomorphisms are given by restriction of G-invariant functions to X × {e} (resp. X × {ẽ}). In particular, both algebras are finitely generated;

(ii) If X has rational singularities, then so do X//G_{e} and X//G_{e}^{o}.

Proof. (i) This is a particular case of [8, 1.2]; see also [15, thm. 4]. What is important here is that k[𝔑] = k[G]^{G_{e}} and k[𝔑̃] = k[G]^{G_{e}^{o}}. (ii) By (1.1)(ii), (1.2), and Boutot’s theorem [1]. □

Let {e_{+}, h, e_{−}} be an 𝔰𝔩_{2}-triple containing e = e_{+} (i.e., [h, e] = 2e, etc.). The subalgebra 𝔰 = ⟨e_{+}, h, e_{−}⟩ is a principal TDS in Kostant’s terminology. Denote by S the corresponding connected subgroup of G. Let V be an arbitrary (finite-dimensional) G-module. By SL_{2}-theory, the eigenvalues of h on V are integral and those on V^{𝔤_{e}} are nonnegative. Hence the action of h yields a ℤ_{≥0}-grading on k[X]^{G_{e}^{o}} (and, of course, on k[X]^{G_{e}}). The n^{th} graded piece is k[X]_{n}^{G_{e}^{o}} = {f ∈ k[X]^{G_{e}^{o}} | h·f = nf} (n = 0, 1, …). Here we regard h as an algebraic vector field on X.

1.4 Proposition.

(1) k[X]_{0}^{G_{e}^{o}} = k[X]_{0}^{G_{e}} = k[X]^{G};

(2) k[X]_{n}^{G_{e}^{o}} is a finite k[X]_{0}^{G_{e}^{o}} -module for each n ∈ ℕ;

(3) k[X]_{n}^{G_{e}} = 0 unless n is even.

Proof. 1. By SL_{2}-theory, any function in k[X]_{0}^{G_{e}^{o}} is S -invariant. It then suffices to prove that the Lie algebra generated by 𝔤_{e} and 𝔰 is equal to 𝔤. This again follows from SL_{2}-theory: successively applying ad e_{−} to a basis of 𝔤_{e} consisting of h-eigenvectors, we may obtain a basis of 𝔤.

2. Associated to h, there is a 1-parameter subgroup (1-PS) ν : k^{∗} → S ⊂ G such that dν(1) = h. Because ν(k^{∗}) normalizes G_{e}, it acts on k[X]^{G_{e}^{o}} and k[X]_{n}^{G_{e}^{o}} = {f ∈ k[X]^{G_{e}^{o}} | ν(t)·f = t^{n}f}. So, (2) becomes a standard fact on covariants of reductive groups.

3. For irreducible V , V^{G_{e}} ≠ 0 if and only if Z(G) acts trivially on V . Since the centre of S lies in Z(G), V |_{S} is the sum of odd-dimensional SL_{2}-modules in this case. Thus, all the eigenvalues of h are even. □

Remark. If X = G and the G-action is given by right multiplication, then k[G]^{G_{e}} = k[𝔑] possesses two gradings. The first one is as above (h-grading) and the second one is inherited from k[𝔤]. The relation between them has been observed in [6, 2.4]: (2n)^{th} piece in h-grading is equal to n^{th} piece in k[𝔤]-grading. In our setting, this easily follows from (1.3)(i).

Interest in k[X]^{G_{e}} is also explained by the fact that k[X]^{T}, where T is a maximal torus, is a deformation of it. This is implicit in [4]. Our next goal is to present accurately this relationship. First, we need to match T and G_{e}. Having fixed e ∈ 𝔑 and hence G_{e}, we set T = Z_{G}(h).

1.5 Theorem. Let X be an arbitrary affine G-variety. Then there is an ascending filtration J = {J^{n}} in k[X]^{T} such that gr_{J}k[X]^{T} ≃ k[X]^{G_{e}}.

Proof. Set J^{−1} = 0 and J^{n} = {x ∈ k[X]^{T} | e^{n+1} · x = 0} for n = 0, 1 . . .. This is well-defined, since each x belong to a finite-dimensional G-submodule and therefore e is a locally-nilpotent endomorphism of k[X]. Then gr_{J}k[X]^{T} := ⊕_{n≥0}J^{n}/J^{n−1}. Define the linear map ψ : gr_{J}k[X]^{T} → k[X] by: \overline{x} ∈ J^{n}/J^{n−1} ↦ ψ(\overline{x}) = \frac{1}{n!}e^{n} · x. From [4, 2.7] it follows that the image of ψ is k[X]^{G_{e}} and ψ is injective. Because e acts as a derivation of k[X], an application of the Leibniz rule shows ψ is an algebra homomorphism. □

The theorem implies that k[X]^{T} is a deformation of k[X]^{G_{e}} in a very precise sense. Practically, this means whenever k[X]^{G_{e}} has a property ‘of open type’, k[X]^{T} also has this property. See [15, §5,6] for a thorough treatment of deformations (contractions) and properties of open type.

1.6 Remark. G/T and G/G_{e} are two representatives of the finite (up to isomorphism) set of regular orbits in 𝔤. More generally, if G/A is an arbitrary regular orbit, then the analogue of (1.5) holds with A in place of T. Actually, all the varieties of the form X//A can be organized as the fibres of an equidimensional map onto an affine space. Endow X × 𝔤 with the diagonal G-action. Associated to the projection ϑ : X × 𝔤 → 𝔤, there is the map of quotient varieties ϑ̄ : (X × 𝔤)//G → 𝔤//G ≃ 𝔸^{rk G}. Given ξ ∈ 𝔤//G, denote by A_{ξ} the stabilizer of a point in the regular G-orbit over ξ. Then ϑ̄^{−1}(ξ) = (X × π_{𝔤}^{−1}(ξ))//G ≃ X//A_{ξ}. The latter isomorphism follows by [8, 1.2], since k[π_{𝔤}^{−1}(ξ)] = k[G/A_{ξ}] [10]. Therefore ϑ̄ is equidimensional, the fibres being of dimension dim X − rk G. Thus, ϑ̄ is flat whenever (X × 𝔤)//G is Cohen-Macaulay. A sufficient condition for it is that X has rational singularities.

2 The null-cone for G_{e}-invariants

Suppose X is an affine G-variety having a fixed point, say x_{0}. For a subgroup H ⊂ G, we set

𝔑_{H}(X) = {x ∈ X | f(x) = f(x_{0}) for any f ∈ k[X]^{H}} .

If k[X]^{H} is finitely generated, then one may think of 𝔑_{H}(X) as the (reduced) fibre containing x_{0} of the quotient map π_{X,H} : X → X//H. By analogy with reductive group actions, we shall say that 𝔑_{H}(X) is the null-cone. However, it fails to be true in general that x belongs to 𝔑_{H}(X) if and only if the closure of Hx contains x_{0}; for instance, if H is unipotent then all H -orbits are closed. The Hilbert-Mumford (=HM) criterion provides a description of 𝔑_{H}(X) in terms of 1-PS whenever H is reductive. Because this does not appeal to invariants, it is helpful to have a similar description in the nonreductive case. As a sample, we mention Brion’s theorem on the null-cone for a maximal unipotent subgroup [2, IV]. Fortunately, the same approach applies to nilpotent tori as well and the answer in terms of 1-PS turns out to be even better, since no preliminary conjugation is required.

Recall that T = Z_{G}(h). Here is the place to put a bit more notation forth. Let Δ be the set of roots relative to T. The unique Borel subalgebra 𝔟 containing e (and 𝔱) determines the subset of positive roots Δ_{+}. Let α_{1}, …, α_{p} (p = rk G) be the corresponding simple roots. Then one may choose corresponding root vectors e_{α_{i}} (i = 1, …, p) in such a way that e = e_{α_{1}} +…+e_{α_{p}}. The fundamental weights are denoted by φ_{1}, …, φ_{p}. In the next theorem, ( | ) stands for the natural pairing between the characters and the 1-parameter subgroups of T. If μ is a character and γ : k^{∗} → T is a 1-PS, then (μ ∘ γ)(t) =: t^{(μ|γ)}, t ∈ k^{∗}.

Note that 𝔑_{G_{e}^{o}}(X) = 𝔑_{G_{e}}(X), because |G_{e} : G_{e}^{o}| < ∞.

2.1 Theorem. Notation as above, the following conditions are equivalent for x ∈ X: (i) x ∈ 𝔑_{G_{e}}(X); (ii) there is a one-parameter subgroup τ : k^{∗} → T such that (α_{i}|τ) > 0 (i = 1, …, p)

and lim_{t→0} τ(t) · x = x_{0}.

Proof. The guiding idea is to consider the G-action on X × 𝔑 in place of the G_{e}-action on X, 𝔑_{G_{e}}(X) being replaced by 𝔑_{G}(X × 𝔑). The latter is determined by the fixed point (x_{0}, 0). The advantage is that one may apply the usual HM-criterion to the Gaction. Later on, a 1-PS τ : k^{∗} → T is said to be strictly dominant whenever (α_{i}|τ) > 0 for i = 1, …, p.

(ii)⇒(i): By (1.3)(i), it is enough to prove that (x, e) ∈ 𝔑_{G}(X×𝔑). Since e = ∑_{i} e_{α_{i}}, lim_{t→0} τ(t) · e = 0. Therefore the closure of G · (x, e) contains (x_{0}, 0) and the assertion follows.

(i)⇒(ii): By (1.3)(i), (x, e) ∈ 𝔑_{G}(X × 𝔑). Therefore, by the HM-criterion, there is a 1-PS γ : k^{∗} → G such that lim_{t→0} γ(t) · (x, e) = (x_{0}, 0).

\underline{Step 1}: Without loss of generality, one may assume that γ(k^{∗}) ⊂ B. For, P := {g ∈ G | lim_{t→0} γ(t)gγ(t)^{−1} exists in G} is a parabolic subgroup. Therefore B ∩ P contains a maximal torus, say T′, of G. Since γ(k^{∗}) is a 1-dimensional torus in P, there is g ∈ P such that g^{−1}γ(k^{∗})g ⊂ T′ ⊂ B. Then having written

g^{−1}γ(t)g · (x, e) = g^{−1}[γ(t)gγ(t)^{−1}]γ(t) · (x, e) ,

we see that this aggregate tends to (x_{0}, 0) as t tends to 0. Thus, one may replace γ by g^{−1}γg.

\underline{Step 2}: Since γ(k^{∗}) ⊂ B, there is u ∈ U (the unipotent radical of B) such that τ(t) := uγ(t)u^{−1} ∈ T for all t ∈ k^{∗}. Hence lim_{t→0} τ(t)u · (x, e) = (x_{0}, 0). In particular, lim_{t→0} τ(t) · (u · e) = 0. Because u · e is still a regular nilpotent element in 𝔲 ⊂ 𝔟, τ must be strictly dominant. Next, we have x_{0} = lim_{t→0} τ(t)u · x = lim_{t→0}[τ(t)uτ(t)^{−1}]τ(t) · x. Since u ∈ U and we already know that τ is strictly dominant, lim_{t→0} τ(t)uτ(t)^{−1} = 1_{G} (the unit in G). Hence, lim_{t→0} τ(t)x = x_{0}, and we are done. □

Denote by W the Weyl group N_{G}(T)/T.

2.2 Corollary.

(i) 𝔑_{G_{e}}(X) ⊂ 𝔑_{T} (X) and 𝔑_{G_{e}}(X) is B -stable; (ii) 𝔑_{T} (X) = W · 𝔑_{G_{e}}(X). In particular, dim 𝔑_{T} (X) = dim 𝔑_{G_{e}}(X).

Proof. (i) Obvious. (ii) By the first part, notation W · 𝔑_{G_{e}}(X) makes sense. Let us say a 1-PS γ is regular if (α|γ) ≠ 0 for each α ∈ Δ. Obviously, each regular 1-PS is W -conjugate to a strictly dominant one. It remains to observe that if x ∈ 𝔑_{T} (X), i.e., lim_{t→0} γ(t) · x = x_{0} for some 1-PS γ, then there is a regular 1-PS γ′ with the same property. □

The corollary yields the following recipe: Consider the set of irreducible components of 𝔑_{T} (X). The group W acts on it. Choose the ‘dominant’ representative in each W -orbit. The resulting set is precisely the set of irreducible components of 𝔑_{G_{e}}(X).

Another curious consequence of the properties of the null-cone is that the fibre of π_{X,G_{e}} containing x_{0} is never reduced (in the scheme-theoretic sense). For the quotient map π_{X,G_{e}} : X → X//G_{e}, denote by 𝔪̄_{0,X} the maximal ideal of the point π_{X,G_{e}}(x_{0}). That is, 𝔪̄_{0,X} = {f ∈ k[X]^{G_{e}} | f(x_{0}) = 0}.

2.3 Theorem. The ideal k[X]𝔪̄_{0,X} ⊂ k[X] is not equal to the radical Rad (k[X]𝔪̄_{0,X}).

Proof. The idea is quite simple. By (2.2)(i), any f ∈ k[X]^{T} such that f(x_{0}) = 0 lies in Rad (k[X]𝔪̄_{0,X}). On the other side, theorem (1.5) suggests it is unlikely that all such functions lie in k[X]𝔪̄_{0,X}. Implementation of this program is easy, if k[X] is graded. Fortunately, the problem can always be reduced to this case. Let O be a minimal G-orbit containing x_{0} in its closure, O ≠ {x_{0}}. Then the stabilizer of a (any) point in O contains a maximal unipotent subgroup of G. (The proof of [3, Lemme on p.401] applies. The only difference in the assumptions is that we do not assume that X can be embedded in an irreducible G-module.) Such varieties have been studied in [16]. If C := \overline{O} is normal, then it is so-called HV-variety of G. In any case, k[C] has a ℤ_{≥0}-grading such that the sum of all pieces of positive grade is the ideal of x_{0} ∈ C. Let f ∈ k[C]^{T} be an invariant of the least positive degree, say n. Since k[C]^{T} ∩ k[C]^{G_{e}} = k[C]^{G} = k, we have f ∉ k[C]^{G_{e}}. It follows from (1.5) that the least positive degree of G_{e}-invariants is also n. Hence f ∉ k[C]𝔪̄_{0,C}. Now, we can extend f to a T -invariant function f̂ on X. (This is possible, for T is reductive.) Obviously, then f̂ ∉ k[X]𝔪̄_{0,X}. □

3 Actions of nilpotent tori on HV-varieties

This section can be viewed as a supplement to my paper [14], where T -actions on HVvarieties were studied. Recall that an HV-variety C(λ) is the closure of the orbit of highest weight vectors in the irreducible G-module V_{λ} with highest weight λ. Interest in G_{e}-invariants on HV-varieties is explained by their relations to Lusztig’s q-analogs of weight multiplicities (see [13, §5]).

First, we say some words on description of the null-cone of G_{e} on HV-varieties. Because C(λ) consists of finitely many B -orbits and 𝔑_{G_{e}}(C(λ)) is B -stable, we need only to indicate a finite collection of points generating these B -orbits. Let v_{λ} ∈ V_{λ} be a highest weight vector. ¿From the Bruhat decomposition it then follows that C(λ) = ( ⨆_{w∈W^{λ}} B · v_{wλ}) ∪ {0}, where W^{λ} is the set of representatives of minimal lehgth

for the coset space W/W_{λ}. In this case dim B · v_{wλ} = l(w), the length of w ∈ W^{λ}. The following is an immediate consequence of (2.1).

3.1 Lemma. v_{wλ} ∈ 𝔑_{G_{e}}(C(λ)) if and only if (wλ, φ_{i}) > 0 for some i ∈ [1, p] if and only if wλ ∉ ∑_{i=1}^{p} ℚ_{≥0}(−α_{i}). □

3.2 Example. Suppose λ = Λ, the highest root. To be in consent with preceding notation, root vectors in 𝔤 are denoted by e_{α}, α ∈ Δ. Here the condition of the lemma reduces to the following: e_{wΛ} ∈ 𝔑_{G_{e}}(C(Λ)) if and only if wΛ ∈ Δ_{+}. It is easy to see that the maximal B -orbits in the null-cone correspond to the simple roots of the form wΛ. That is, the number of irreducible components of 𝔑_{G_{e}}(C(Λ)) is equal to the number of long simple roots. Furthermore, an easy computation shows (if a simple root α_{i} is long !)

dim B · e_{α_{i}} = \frac{1}{2}(#\{α ∈ Δ | (α, α_{i}) > 0\} + 1) = \frac{1}{2} dim C(Λ) .

Thus, dim 𝔑_{G_{e}}(C(Λ)) = \frac{1}{2} dim C(Λ). (It also follows from the above discussion that 𝔑_{G_{e}}(C(Λ)) = C(Λ) ∩ 𝔲.) Making use of it, one can prove that dim C(Λ) ≥ 2rk G and the equality holds if and only if the quotient map π_{C(Λ),G_{e}} is equidimensional:

dim C(Λ) ≤ dim C(Λ)//G_{e} + dim 𝔑_{G_{e}}(C(Λ)) = dim C(Λ) − rk G + \frac{1}{2} dim C(Λ).

Since k[C(λ)]^{T} is a deformation of k[C(λ)]^{G_{e}}, it follows that properties of the latter are in general worse. However, as we shall see, no change for the worse actually happens for the property of algebras of being polynomial (if G is simple).

3.3 Theorem. Suppose G is a simple algebraic group. Then k[C(λ)]^{G_{e}} is polynomial if and only if k[C(λ)]^{T} is.

Proof. The “only if” part follows by standard deformation-type argument. The real job is required for the “if”-part. Unfortunately, the proof is case-by-case. The table of all λ’s such that C(λ)//T is an affine space is found in [14, 5.3]. It then suffices to prove that C(λ)//G_{e} is an affine space in all these cases. Fortunately, this will be derived from some general results on algebras of G_{e}-invariants when G = A_{p}. The dominant weights in [loc. cit.] break into the two parts. The first part includes the weights (and some of their multiples) corresponding to the simplest representations of the classical groups. The fact that k[C(λ)]^{G_{e}} is polynomial for them is proved in [13, 5.9]. The second part consists— up to the symmetry of Dynkin diagram—of two infinite series of weights for G = A_{p}: φ_{1} + φ_{p} + n(p + 1)φ_{1}, (p − 1)φ_{1} + φ_{2} + n(p + 1)φ_{1}, n = 0, 1, . . .. Here is the job we are to do. Actually, we shall show there is a vast area in the Weyl chamber, including these weights, such that C(λ)//T and C(λ)//G_{e} are (noncanonically) isomorphic, see (3.7). □

Before switching to the A_{p}-case, we state an easy general result on representations:

3.4 Proposition. Let P be a parabolic subgroup in G with the unipotent radical N. Let P_{−} be an opposite parabolic subgroup. Let V be an irreducible G-module. Then there is a unique P_{−}-stable complement to V^{N}. Furthermore, this complement is the unique maximal proper P_{−}-submodule of V . In particular, the P_{−}-equivariant projection pr_{N} : V → V^{N} is well-defined. (Hint: consider V as a P ∩ P_{−}-module and take the sum of all submodules distinct from V^{N}, cf. [14, 2.3].) □

Until the end of section, G = A_{p} = SL_{p+1} with the obvious choices of B —as the uppertriangular matrices—and T —as the diagonal matrices. Recall that e = e_{α_{1}} + … + e_{α_{p}}. Let L be the simple subgroup of G whose simple roots are α_{2}, …, α_{p}. Denote by P the parabolic subgroup containing B whose semisimple part is L. The unipotent radical N is p-dimensional, it corresponds to the roots α_{1}, α_{1} + α_{2}, …, α_{1} + … + α_{p}.^{1} Define P_{−} to be the standard opposite to P parabolic subgroup.

1This is the only case, among all simple groups, where dim N = rk G. To a great extent, the following results are explained by this equality.

3.5 Proposition. 1. 𝔭_{−} ⊕ 𝔤_{e} = 𝔤 and hence P_{−}G_{e}^{o} is dense in G;

2. Let V be an irreducible G-module. Then the restriction onto V^{G_{e}^{o}} of the P_{−}- equivariant projection pr_{N} : V → V^{N} is injective. In particular, dim V^{G_{e}^{o}} ≤ dim V^{N}.

Proof. 1. Straightforward.

2. Denote the vector pr_{N}(v) by v_{N}. Let v ∈ V^{G_{e}^{o}}. Then V = ⟨G · v⟩ = ⟨P_{−}G_{e}^{o} · v⟩ = ⟨P_{−} · v⟩. That is, v does not belong to a P_{−}-stable subspace of V . By (3.4), it then forces v_{N} ≠ 0. □

Recall that k[C(λ)] is graded, the piece of grade n being V_{nλ^{∗}}. Here λ^{∗} stands for the highest weight of the dual G-module. While dealing with C(λ) and k[C(λ)], it is unavoidable to get somewhere an abundance of ‘*’ . To minimize the total amount, the variety will frequently be denoted as C(λ^{∗}).

3.6 Theorem. For any dominant weight λ, k[C(λ^{∗})]^{G_{e}^{o}} is isomorphic to a graded subalgebra of k[C(λ^{∗})]^{N}.

Proof. Subalgebras of k[C(λ^{∗})] are assumed to have the inherited grading. For each n, we have the injective linear map v ∈ (V_{nλ})^{G_{e}^{o}} ↦ v_{N} ∈ (V_{nλ})^{N}. Putting them together, we obtain an injective linear map of algebras of regular functions. Let us check that this map respects multiplication. Take v_{i} ∈ (V_{n_{i}λ})^{G_{e}^{o}}, i = 1, 2. Write v_{i} = (v_{i})_{N} + ṽ_{i}, where ṽ_{i} lies in the maximal proper P_{−}-submodule of V_{n_{i}λ}. Then v_{1}v_{2} = (v_{1})_{N}(v_{2})_{N} + ṽ and it is easy to see that ṽ belongs to a proper P_{−}-submodule of V_{(n_{1}+n_{2})λ}. ¿From (3.4) it then follows that (v_{1}v_{2})_{N} = (v_{1})_{N}(v_{2})_{N}. □

Thus, whenever one knows that—for some reason—dimensions of spaces (V_{nλ})^{G_{e}^{o}} and (V_{nλ})^{N} are equal for all n ∈ ℕ, it will imply the corresponding algebras of invariants on C(λ^{∗}) are isomorphic. Surprisingly enough, this possibility is not an imaginary one. Recall from [14, sect. 4] the following condition on a dominant weight λ = ∑_{i=1}^{p} l_{i}φ_{i}:

(◇) l_{1} ≥ (p − 1)l_{2} + (p − 2)l_{3} + … + l_{p} .

3.7 Theorem. Let λ = ∑_{i=1}^{p} l_{i}φ_{i} be a dominant weight of SL_{p+1} satisfying (◇). Then

(i) k[C(λ^{∗})]^{G_{e}^{o}} ≃ k[C(λ^{∗})]^{N}; (ii) k[C(λ^{∗})]^{T} ≃ k[C(λ^{∗})]^{G_{e}}.

Proof. (i) We are to show dim(V_{nλ})^{G_{e}^{o}} = dim(V_{nλ})^{N} for all n. Since this relies on inequality (◇), which is linear, it suffices to handle the case n = 1. By [14, 2.2], (V_{λ})^{N} is an irreducible L-module whose highest weight λ′ is obtained from λ by a truncation. That is, dim(V_{λ})^{N} = dim V_{λ′}. This is true regardless of (◇). By a result of W. Graham [7, 1.6], dim(V_{λ})^{G_{e}^{o}} is dimension of the largest weight space in V_{λ}. This space corresponds to the unique minuscule dominant weight that is a weight of V_{λ}. For A_{p}, the muniscule weights are φ_{0} = 0 and the fundamental weights φ_{i}, i = 1, …, p. A straightforward calculation shows that condition “φ_{i} is a weight of V_{λ}” is equivalent to “l_{1} − (p − 1)l_{2} − (p − 2)l_{3} − … − l_{p} ≡ i mod (p + 1)”. That is, l_{1} = (p − 1)l_{2} + … + l_{p} + i + m(p + 1). Then (◇) means m ≥ 0. Lemma 4.1 in [14] says that, under a certain constraint on weight μ, dim V_{λ}^{μ} = dim V_{λ′}. We shall prove this applies to μ = φ_{i}. That constraint has to do with presentation of weights in terms of partitions. Namely, if \underline{λ} = (λ_{1}, …, λ_{p}, 0) is the partition associated to λ, then the condition is: each part of the partition associated to φ_{i} must be ≥ λ_{2}. By definition, λ_{2} = l_{2} + … + l_{p} and the above formula for l_{1} shows that the partition corresponding to φ_{i} has i parts of size l_{2} +…+l_{p} +m+1 and p+1−i parts of size l_{2} + … + l_{p} + m. Thus, dim(V_{λ})^{G_{e}^{o}} = dim V_{λ}^{φ_{i}} = dim V_{λ′} whenever m ≥ 0.

(ii) Both (V_{nλ})^{T} and (V_{nλ})^{G_{e}} are nonzero if and only if nλ lies in the root lattice Q. Hence it is enough to treat the case λ ∈ Q. Then there is no difference between G_{e}- and G_{e}^{o}-invariants, and an isomorphism of k[C(λ^{∗})]^{T} and k[C(λ^{∗})]^{N} follows from [14, 2.3,4.4]. □ Remarks. 1. We have C(λ)//H ≃ C(λ^{∗})//H, if H = T or G_{e} or G_{e}^{o}. But this is not true for N, since P is not conjugate to P_{−}.

2. Coming back to the proof of (3.3), we see that (◇) holds for the two series of weights mentioned therein. This completes that proof.

Examples. 1. For SL_{3}, condition (◇) reads l_{1} ≥ l_{2}. Whence by symmetry of the situation, we conclude C(λ)//T ≃ C(λ)//G_{e} and C(λ)//G_{e}^{o} ≃ C(λ)//N for any λ.

2. In some cases, information on the graded structure of k[C(λ^{∗})]^{G_{e}} allows us to write explicitly down the q-analogs of zero-weight multiplicity in the modules V_{nλ}, n = 1, 2 . . .. Now, we can add (at least) one more example to the collection of formulas in [13, 5.9,5.11]. First, recall some relevant notions. The algebra k[C(λ^{∗})]^{G_{e}} is always (ℤ_{≥0})^{2}-graded: together with the grading coming from k[C(λ^{∗})] it possesses the \frac{1}{2}h-grading. The precise definition is that k[C(λ^{∗})]_{l,n}^{G_{e}} := {f ∈ (V_{nλ})^{G_{e}} | h · f = 2lf} (our present h is not the same as in [13], the transition factor is 1/2). The Poincaré series is defined by

F_{λ}(q, t) = ∑_{l,n≥0} dim k[C(λ^{∗})]_{l,n}^{G_{e}} q^{l}t^{n} .

If we write it as F_{λ}(q, t) = ∑_{n} m_{nλ}^{0}(q)t^{n}, then m_{nλ}^{0}(q) is nothing but Lusztig’s q-analog of zero-weight multiplicity. For SL_{p+1}, these were known before as Kostka-Foulkes polynomials K_{nλ,0}. Take λ = (p − 1)φ_{1} + φ_{2} for SL_{p+1}. We have proved that k[C(λ^{∗})]^{G_{e}} is polynomial. It is generated by functions lying in V_{λ}, i.e., those having t-degree 1 (since this is true for k[C(λ^{∗})]^{T}, see [14]). It is not hard to compute the eigenvalues of \frac{1}{2}h on

(V_{λ})^{G_{e}}. These are \binom{p}{2} + i, i = 0, 1, …, p − 1. Therefore F_{λ}(q, t) = \frac{1}{∏_{i=0}^{p−1}(1 − q^{\binom{p}{2}+i}t)} .

Whence m_{nλ}^{0}(q) = q^{\binom{p}{2}} \left[{p+n−1 \atop p−1}\right], where the brackets stand for a Gaussian polynomial in q.

4 Some applications

We come back to the general case, where X is an affine G-variety. Our applications concern some properties of the quotient map π_{G} : X × 𝔤 → (X × 𝔤)//G, where X × 𝔤 is equipped with the diagonal G-action. Let A be a finitely generated k-algebra. The embedding dimension of A (or Spec A) is the minimal number of generators of A. This will be denoted by ed A (or ed Spec A).

4.1 Theorem. Let X be an affine G-variety. Then ed (X×𝔤)//G ≤ rk G+ed X//G_{e}.

Proof. The polynomial algebra k[𝔤] has the standard ℤ_{≥0}-grading. Give k[X × 𝔤]^{G} a ℤ_{≥0}-grading such that its n^{th} piece is (k[X] ⊗ k[𝔤]_{n})^{G}. Let f_{1}, …, f_{p} (p = rk G) be free homogeneous generators of k[𝔤]^{G} and {ḡ_{1}, …, ḡ_{s}} a minimal homogeneous generating system of k[X × 𝔑]^{G}. Recall that the last algebra is isomorphic to k[X]^{G_{e}}. Lift the ḡ_{i}’s to G-invariant functions on X × 𝔤. That is, take for each i a g_{i} ∈ k[X × 𝔤]^{G} such that g_{i}|_{X×𝔑} = ḡ_{i} and deg g_{i} = deg ḡ_{i}. Note that if deg ḡ_{i} = 0, then ḡ_{i} ∈ k[X]^{G}. In this case g_{i} is uniquely determined as the image of ḡ_{i} under the injection k[X]^{G} → k[X ×𝔤]^{G}. I claim that k[X ×𝔤]^{G} is generated by f_{1}, …, f_{p}, g_{1}, …, g_{s}. The proof is by induction on degree. Let f be a homogeneous G-invariant. If deg f = 0, then f is a polynomial in g_{i}’s such that deg g_{i} = 0. Suppose deg f > 0. Then f̄:= f|_{X×𝔑} = F (ḡ_{1}, …, ḡ_{s}), a polynomial in ḡ_{i}’s. Since (f_{1}, …, f_{p}) = Rad (f_{1}, …, f_{p}) [10], we have f − F (g_{1}, …, g_{s}) = ∑_{i=1}^{p} r_{i}f_{i}. Applying the Reynolds operator, one can get r_{i} ∈ k[X × 𝔤]^{G}. Because deg f_{i} > 0 and then deg r_{i} < deg f, the assertion follows by the induction hypothesis. □

4.2 Corollary. 1. If X//G_{e} ≃ 𝔸^{s}, then (X × 𝔤)//G ≃ 𝔸^{s+rk G};

2. If X//G_{e} is a hypersurface then (X ×𝔤)//G is either an affine space or a hypersurface. □

Examples. 1. Take X = C(λ), an HV-variety. Denote by G_{λ} the stabilizer of v_{λ} ∈ V_{λ}. Because k[C(λ)] = k[G/G_{λ}], we have k[C(λ) × 𝔤]^{G} ≃ k[𝔤]^{G_{λ}}. Thus, the following holds: If k[C(λ)]^{G_{e}} is a polynomial algebra, then so does k[𝔤]^{G_{λ}}. Then consulting with (3.3) and the table in [14, 5.3], one finds some admissible G_{λ}. The list obtained includes all (but is not exhausted by) the examples found by Levstein [11].

2. Let G = SL_{p+1} and X = V_{φ_{1}} ⊕ V_{φ_{p}}. Here X//G_{e} is a hypersurface. Therefore (𝔤 ⊕ V_{φ_{1}} ⊕ V_{φ_{p}})//SL_{p+1} is a hypersurface, for it is not an affine space.

¿From now on, we assume that X has a fixed point x_{0}. Then X × 𝔤 has a fixed point (x_{0}, 0) and we shall compare various null-cones in X and X × 𝔤.

4.3 Theorem.

(i) 𝔑_{G}(X × 𝔤) = G · (𝔑_{G_{e}}(X) × 𝔲);

(ii) The map Y_{i} ↦ G · (Y_{i} × 𝔲) sets up a one-to-one correspondence between the irreducible components of 𝔑_{G_{e}}(X) and 𝔑_{G}(X × 𝔤); dim G · (Y_{i} × 𝔲) = dim Y_{i} + dim 𝔑.

Proof. Note that 𝔑 is the null-cone in 𝔤. Therefore 𝔑_{G}(X × 𝔤) = 𝔑_{G}(X × 𝔑).

(i) By (1.3)(i), 𝔑_{G}(X × 𝔑) ∩ (X × {e}) = 𝔑_{G_{e}}(X) × {e}. Since 𝔑_{G_{e}}(X) is B -stable, we have \overline{B · (𝔑_{G_{e}}(X) × \{e\})} = 𝔑_{G_{e}}(X) × 𝔲. This proves inclusion “⊃”. The proof of the converse uses the HM-criterion. If (x, n) ∈ 𝔑_{G}(X ×𝔑), then there are a 1-PS γ : k^{∗} → T and g ∈ G such that lim_{t→0} γ(t)g · (x, n) = (x_{0}, 0). Without loss of generality, one may assume that γ is regular. Conjugating the point, one may assume that g = 1_{G} and then that γ is strictly dominant. In the last case, the condition lim_{t→0} γ(t) · (x, n) = (x_{0}, 0) implies x ∈ 𝔑_{G_{e}}(X) and n ∈ 𝔲. (ii) Let 𝔑_{G_{e}}(X) = ∪_{i}Y_{i} be the irreducible decomposition. Then each Z_{i} := G · (Y_{i} × 𝔲) is a closed subvariety of 𝔑_{G}(X × 𝔑). Because Z_{i} ∩ (X × {e}) = Y_{i} × {e}, we see that Z_{i}’s are the irreducible components of 𝔑_{G}(X × 𝔑). The dimension formula is obvious. □

Next, I am going to prove that 𝔑_{G}(X ×𝔤) is never reduced in the scheme-theoretic sense. It will be based on (2.3) and the following technical assertion:

4.4 Theorem. Suppose X is normal and f ∈ k[X] vanishes on 𝔑_{G_{e}}(X). Then there exists F ∈ k[X × 𝔑] such that F |_{X×{e}} = f and F |_{𝔑_{G}(X×𝔑)} = 0.

Proof. First, extend f to a function f̂ on X × 𝔲 by f̂(x, n) := f(x). Then f̂|_{𝔑_{Ge}(X)×𝔲} = 0. Let U_{−} be the opposite to U maximal unipotent subgroup, i.e., U_{−}B is the big cell in G. Consider the map δ : U_{−} × X × 𝔲 → X × 𝔑, (u, x, n) ↦ (u · x, (Ad u)n). Since 𝔑 = G𝔲 = \overline{U_{−}𝔲}, we have δ is dominant. Moreover, δ is birational. It follows from the easy fact that if e′ ∈ 𝔲 is regular and (Ad u)e′ ∈ 𝔲 for u ∈ U_{−}, then u = 1_{G}. Hence each polynomial function on U_{−} × X × 𝔲 gives rise to a rational function on X × 𝔑. Define f̆ ∈ k[U_{−} × X × 𝔲] by f̆(u, x, n) := f̂(x, n). If f̆ turns out to be polynomial on X × 𝔑, then F := f̆ and we are done. (The equality 𝔑_{G}(X × 𝔑) = \overline{U_{−}(𝔑_{G_{e}}(X) × 𝔲)} guarantee us that F vanishes on 𝔑_{G}(X × 𝔑).) If not, then f̆ has the divisor of poles, which we denote by (f̆)_{∞}. The map δ is an isomorphism over the open subset X×U_{−}B·e ⊂ X×𝔑. Hence Supp (f̆)_{∞} ⊂ X × D, where D = 𝔑 \ U_{−}B · e. Let g ∈ k[𝔑] be a function such that g(e) = 1 and Supp (g)_{0} ⊃ D. (Actually, there is g such that Supp (g)_{0} = D, for the divisor class group Cl (𝔑) is finite.) Then f̆g^{l} ∈ k[X × 𝔑] for some l > 0 and we may take F = f̆g^{l}. □

The theorem can be restated as follows: Let I_{1} (resp. I_{2}) be the ideal of all functions in k[X × 𝔑] vanishing on X × {e} (resp. 𝔑_{G}(X × 𝔑)). Then I_{1} + I_{2} = Rad (I_{1} + I_{2}).

4.5 Theorem. Let X be a normal G-variety having a fixed point x_{0}. Then the ideal in k[X × 𝔤] (or in k[X × 𝔑]) generated by all G-invariant functions vanishing at (x_{0}, 0) is not equal to its radical.

Proof. By (2.3), there is f ∈ k[X] such that f ∉ k[X]𝔪̄_{0,X} and f vanishes on 𝔑_{G_{e}}(X). Let F be an extension of f to X × 𝔑 given by (4.4). It then follows from (1.3)(i) that F does not belong to the ideal in k[X × 𝔑] generated by G-invariants vanishing at (x_{0}, 0).

Extending somehow F to a function F̂ on X × 𝔤, we obtain the required function in k[X × 𝔤]. □

Recall that X is said to be conical, if k[X] is ℤ_{≥0}-graded and the zero-grade piece is just k. The point corresponding to the maximal homogeneous ideal is called the vertex. It is clear what is called a conical G-variety. If X is a conical G-variety, then X × 𝔤 has a conical structure, too.

4.6 Proposition. X being a conical G-variety, the following conditions are equivalent: (i) π_{X,T} : X → X//T is equidimensional; (ii) π_{X,G_{e}} : X → X//G_{e} is equidimensional; (iii) π_{G} : X × 𝔤 → (X × 𝔤)//G is equidimensional.

Proof. In the conical case, the fibre of the quotient map containing the vertex, which is a fixed point, is of maximal dimension. Then equivalence of (i) and (ii) follows from (2.2)(ii). Equivalence of (ii) and (iii) follows from (4.3)(ii) by straightforward calculations of dimensions involved. □

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ul. Akad. Anokhina, d.30, kor.1, kv.7 Moscow 117602 Russia Current address: Max-Planck-Institut für Mathematik

Gottfried-Claren-Str. 26 D-53225 Bonn, Deutschland

e-mail: panyush@dpa.msk.ru panyush@mpim-bonn.mpg.de