Birational sheets in reductive groups

We define the group analogue of birational sheets, a construction performed by Losev for reductive Lie algebras. For G semisimple simply connected, we describe birational sheets in terms of Lusztig-Spaltenstein induction and we prove that they form a partition of G, and that they are unibranch varieties with smooth normalization by means of a local study.


Introduction
The action of a connected algebraic group G on an algebraic variety X can be studied by gathering orbits in finitely many families to deduce properties shared by orbits in the same collection. One way of grouping orbits together is to form sheets, i.e. maximal irreducibile subsets of X consisting of equidimensional orbits. In [4], Borho and Kraft studied sheets for the adjoint action of a semisimple connected group G on its Lie algebra g: the authors considered non-nilpotent orbits as deformations of nilpotent ones of the same dimension to compare the G-module structure of their ring of regular functions. In the same paper, sheets and their closures were described set-theoretically as unions of decomposition classes. The latter form a partition of g into finitely-many, irreducible, smooth, G-stable, locally closed subsets, see [8,9]. In [3], Borho described sheets of g in terms of Lusztig-Spaltenstein induction. If g = sl n (C), sheets are parametrized by partitions of n and any two distinct sheets have trivial intersection. This does not hold in general: for example, all simple non-simply laced Lie algebras present two sheets of subregular elements intersecting non-trivially. For g simple and classical, all sheets are smooth (see [18]), but this does not extend to exceptional Lie algebras (the list of smooth sheets is to appear in [10]). In [23, §4], Losev applied the theory of universal Poisson deformations of conical symplectic singularities to define birational sheets of g. He proved that, unlike sheets, birational sheets form a partition of g, they are smooth up to a bijective normalization and all birational sheets of g simple and classical are smooth. Furthermore, the G-module structure of the ring of functions of adjoint orbits in the same birational sheet is preserved and Losev conjectures that birational sheets can be parametrized by this invariant. The group analogue of decomposition classes, called Jordan classes, first appeared in Lusztig's paper [24]: they provide the stratification with respect to which character sheaves are constructible. Properties of such objects and of their closures were studied in [12] to describe sheets for the conjugacy action of a reductive group G on itself.
In this work we define a group analogue of Losev's birational sheets. Motivation behind the study of this problem is its connection with Representation Theory; in particular, we are interested in the aforementioned Losev's conjecture: a group analogue will be object of study in a forthcoming project. After introducing some notation, in Section 3 we collect and reorganize existing results on induction of conjugacy classes. Induction of unipotent classes was defined by Lusztig and Spaltenstein in [26] and it was then generalized to a non-unipotent conjugacy class in [12] readapting arguments of [3] to the group case. Following this approach and inspired by [23], we define birational induction and weakly birational induction of a conjugacy class requiring birationality of two related maps and we compare these two notions. The last part of Section 3 is devoted to extending properties enjoyed by induction to the case of birational induction. In particular, Lemma 3.14 states a criterion which gives a sufficient condition for a unipotent conjugacy class to be birationally induced. The first main result of the work is Theorem 4.5 in Section 4, where we prove that any conjugacy class of G can be weakly birationally induced in a unique way up to G-conjugacy under some minimality conditions on the data needed to define induction. Section 5 recalls some notions on Jordan classes in G. We restrict to the case G semisimple and simply connected to define and describe the birational closure of a Jordan class. Subsequently we define birational sheets: in Theorem 5.20 we prove that they form a partition of the group. We proceed to compare birational sheets with sheets from a structural point of view. The section ends with some remarks on a possible generalization of the results for G not necessarily simply connected, obtained by relaxing the requirements on the induction. Finally, Section 6 analyzes the local geometry of the birational closure of a Jordan class under the assumption that G is semisimple and simply connected, using results from [1]. As an application, for G semisimple simply connected, we show that birational sheets are unibranch with smooth normalization and that birational sheets are smooth in the classical case (Theorems 6.7 and 6.8).

Notations and conventions
Let G be a complex connected reductive linear algebraic group and let g := Lie(G) be its Lie algebra. For an algebraic subgroup K ≤ G, we denote by K • its identity component, by Z(K) its centre, and by Aut(K) the set of its automorphisms as an algebraic group. If X is a K-set, X/K denotes the set of K-orbits. When K ≤ G acts regularly on a variety X and x ∈ X, the K-orbit of x is denoted by K · x or O X x . For any n ∈ N, we define the locally closed subsets When we consider the conjugacy (resp. the adjoint) action of G on itself (resp. on g) we adopt the following notation for stabilizers, very common in the literature. When G acts on X = G via conjugation, for g ∈ G, we write C G (g) : . When G acts on X = g via the adjoint action, for η ∈ g, we write C G (η) : We define the centralizer of a Lie subalgebra k ⊂ g as c g (k) := {ξ ∈ g | [η, ξ] = 0 for all η ∈ k}.
Let T be a fixed maximal torus in G, B a fixed Borel subgroup of G containing T . We set g := Lie(G), h := Lie(T ), b := Lie(B). The symbol P (resp. p) denotes a parabolic subgroup of G (resp. a parabolic subalgebra of g). A Levi subgroup L ≤ G (resp. a Levi subalgebra l ⊂ g) is a Levi factor of a parabolic P ≤ G (resp. p ⊂ g).
We denote by W the Weyl group of G, by Φ the root system associated to T , by Φ + the set of positive roots relative to B and by ∆ the base for Φ extracted from Φ + . For each α ∈ Φ, we denote by U α the corresponding root subgroup in G. When Φ is irreducible, we write ∆ = {α i | i = 1, 2, . . . , n} following the numbering in [6, Planches I-IX] and −α 0 for the highest root with respect to ∆. We set∆ = ∆ ∪ {α 0 }. A standard parabolic subgroup is P ≤ G such that B ≤ P : then there is Θ ⊂ ∆ such that P = P Θ := L Θ U Θ , where L Θ := T, U α , U −α | α ∈ Θ is called a standard Levi subgroup and U Θ = α∈Φ + \ZΦ U α is the unipotent radical of P Θ . Standard parabolic (resp. standard Levi) subalgebras are the Lie algebras of standard parabolic (resp. standard Levi) subgroups.
For s ∈ G semisimple, C G (s) • is called a pseudo-Levi subgroup, following [28]. If Φ is irreducible, any pseudo-Levi subgroup of G is conjugate to a standard pseudo-Levi group M Θ := T, U α , U −α | α ∈ Θ for some Θ ∆ , [28,Proposition 3]. Let M ≤ G be a pseudo-Levi and let Z = Z(M ). For z ∈ Z, we say that the connected component Z • z ⊂ Z satisfies the regularity If K ≤ G is connected reductive and k := Lie(K), we write U K for the unipotent variety of K and N k for the nilpotent cone of k; we also set U := U G and N := N g . The set of all K-conjugacy classes of K is denoted K/K. A central isogeny π : K → K is a surjective group homomorphism with ker π ≤ Z(K).
In the examples, we will use the following conventions. For n ∈ N \ {0}, let J ′ n be the square matrix of order n whose elements on the antidiagonal are 1 and all other entries are 0. We denote by J 2n := 0 J ′ n −J ′ n 0 , and we realize the symplectic group as Sp 2n (C) = {A ∈ GL 2n (C) | A T J 2n A = J 2n }. As fixed Borel we choose the subgroup of upper-triangular matrices in Sp 2n (C) and as fixed torus we select the subgroup of diagonal matrices in Sp 2n (C).

Birationality of the generalized Springer map
Let H ≤ G be closed and let X be an irreducible H-variety, then H acts on G × X via h · (g, x) = (gh −1 , h · x) for h ∈ H, g ∈ G, x ∈ X. The orbit set (G × X)/H is denoted by G × H X and it is endowed with the structure of an irreducible variety of dimension dim G/H + dim X; we write g * x for the class of (g, x) ∈ G × X. For g ′ ∈ G and g * x ∈ G × H X, we have a G-action on There is a one-to-one correspondence between H-stable subsets of X and G-stable subsets of G × H X assigning the orbit H · x to the orbit G · (1 * x); we also have G g * x = gH x g −1 , for all g ∈ G, x ∈ X.
Suppose in addition X is a subvariety of a G-stable variety Y and that the H-action on X is the restriction of the G-action on Y . Then we define a surjective G-equivariant morphism For x ∈ X, we have (see [4, Proof of Lemma 7.10]): Lemma 3.1. Let P ≤ G be parabolic and let X be a closed P -subvariety of the G-variety Y . Let γ : G × P X → G · X be the map g * x → g · x. Assume: Then: Proof. We want to make use of [19,Lemma 8.8]. Observe first that G × P X identifies with a locally closed subvariety of G/P × Y via the closed embedding g * x → (gP, g · x) (see [4, §7.9]). Jantzen's proof still holds substituting a P -submodule of a G-module with the closed P -stable variety X inside the G-stable variety Y . Moreover, if ϕ x : G → O G x is the orbit map, the condition Lie(G x ) = ker(d 1 ϕ x ) is always fulfilled when the base field is C (see [19, Remark on Lemma 8.8]). For these reasons, to prove our result, we can proceed as in [19,Lemma 8.8], provided that we show that its assumptions are satisfied, namely that there exists x ∈ X with P · x = X and G • x ≤ P . By [29,Theorem 5.1.6], there exists a non-empty open U ⊂ G · X whose fibres through γ are finite: Again by (H1), the set X meets O non-trivially. By (1), for x ∈ O ∩X, we have G • x ≤ P and O ∩X is a union of finitely many P -orbits. Since X is irreducible and O ∩X is open in O ∩ X = X by (H1), we have X = O ∩X = P · x, for some x ∈ O ∩X.
Let P = LU be the Levi decomposition of P . Let O L ∈ L/L and specialize the above construction to the case X = O L U , Y = G, where P acts on X by conjugacy. The generalized Springer map is: The image of γ is the closure of a single conjugacy class O G ∈ G/G, called the conjugacy class induced from (L, O L ), see [12,26]. The definition of induced conjugacy class only depends on the Levi L ≤ G and on the class O L and not on the parabolic P containing L. For any Levi subgroup L ≤ G, we introduce notation Ind G L : L/L → G/G associating to each class O L ∈ L/L the induced conjugacy class Ind G [12,Proposition 4.6]). A unipotent conjugacy class O G is rigid in G if it cannot be induced by a proper Levi subgroup L G and a unipotent class O L ∈ U L /L.
The case of birational induction of adjoint orbits has been studied in [8,15,23,27]. Here we deal with birationality of γ for induction of conjugacy classes. (i) γ is birational; Proof. We want to apply Lemma 3.1 with P acting by conjugacy on

Reduction to the unipotent case
We will make use of the following result: The last statement follows directly.
In the remainder of this section, we recall how induction of a conjugacy class of G is related to induction of a unipotent class in a pseudo-Levi subgroup of G.
Let L ≤ G be a Levi subgroup and let P = LU ≤ G be a parabolic with Levi factor L and let su ∈ L. It was proven in [12, Proposition 4.6] that: Induction is transitive, i.e. if M ≤ G is a Levi subgroup, L is a Levi subgroup of M and O L su ∈ L/L, then: where we used [26, §1.7].
We can assume that T ≤ L and s ∈ T .
We compare the two morphisms: Lemma 3.5. Let γ and γ s be as in (3) and (4), respectively. Set Proof. We will prove birationality verifying condition (iv) of Lemma 3.2.
The following example shows that in general, the birationality of γ s does not imply the birationality of γ.
On the other hand, γ as in (3) is not birational, because C G (sv) ≤ P , for example

Induction of adjoint orbits in a Lie algebra
In this section we describe birational induction in the case of the adjoint action of G on its Lie algebra g. We follow the approach of [23, §1.4 and §4]: we alert the reader that in [23] most results are formulated in terms of G-coadjoint orbits of g * . It is possible to translate all the statements in terms of G-adjoint orbits of g after choosing a G-equivariant non-degenerate symmetric associative bilinear form on g (see [7, §4, Proposition 5]), which yields a G-equivariant isomorphism of vector spaces g ≃ g * .
For a Levi subalgebra l ⊂ g, let ζ ∈ z(l) and O l ∈ N l /L 1 . Include l as the Levi factor of a parabolic subalgebra p = l + n and let P ≤ G be such that Lie(P ) = p. The group P acts via the adjoint action on (ζ + O l + n) and we have the generalized Springer map: When γ is birational, . Therefore, the hypotheses of Lemma 3.1 are satisfied with X = ζ + O l + n and Y = g. Moreover, an analogue of Lemma 3.2 holds. Since ζ ∈ g is semisimple, C G (ζ) is connected, see [30, §3]. In the setting of (5), we have Lie(C G (ζ)) = c g (ζ) and C P (ζ) = P ∩ C G (ζ) is a parabolic subgroup of C G (ζ); moreover, l is a Levi factor of the parabolic subalgebra p ζ := Lie(C P (ζ)), write n ζ for its nilradical. Consider the generalized Springer map: Remark 3.7. The orbit O g is birationally induced from (l, ζ, O l ) if and only if the nilpotent orbit Ind Proof. Lemma 3.5 still holds with the necessary adjustments, so that γ in (5) is birational if and only if γ ζ in (6) is birational.

Birationality for induction of conjugacy classes
In this section we discuss the definition of birational induction of a class O G sv ∈ G/G. (3) is birational; 1 We remark that induction in g can be defined for the adjoint orbit of any element σ + ν in a Levi subalgebra l ⊂ g, see [3, §2]. The reduction to the adopted definition can be obtained from [3, Satz 2.1, 3. Fall].
(b) weakly birationally induced from (L, O L su ) if the generalized Springer map then it is also weakly birationally induced from (L, O L su ). Moreover, the two notions coincide when G is semisimple simply connected or when O G = z O G u for z ∈ Z(G) and u ∈ U (this is the case if and only if the inducing orbit is z O L with O L ∈ U L /L). For this reason, in such cases, we will always omit the adverb "weakly".
We may drop one, or both of the elements of the pair of inducing data (L, O L su ) in the notation when they are clear from the context or they are not relevant. In particular, we will say that the class O G ∈ G/G is (non-trivially) birationally induced (resp. weakly birationally induced) if there exists a proper Levi subgroup L G and a conjugacy class O L ∈ L/L such that O G is birationally induced (resp. weakly birationally induced) from (L, O L ).
In the following, we focus on induction of unipotent classes and we show that most properties carry over to the birational case.

Interaction with isogeny
Induction and birational induction behave well with respect to Springer's isomorphism φ : Proof. Set p := Lie(P ), q := Lie(Q) with Levi decompositions p = l+ n p and q = l+ n q , respectively.
. Consider the generalized Springer maps γ p : G × P (O l + n p ) → O g and γ q : G × Q (O l + n q ) → O g . By Remark 3.10, γ P (resp. γ Q ) is birational if and only if γ p (resp. γ q ) is birational. The degrees of γ p and of γ q are the same. This follows from [5, Proof of Corollary 3.9] where a formula for the degree of the Springer generalized map γ p is given in terms of (l, O l ), and these data are independent of the parabolic.

A sufficient condition for birationality
Next result can be used to test if a unipotent class is birationally induced.    Suppose that γ G L is birational, then for lu 1 u 2 ∈ O G as above, we have

Uniqueness of birational induction
In this section we establish an explicit bijection between conjugacy classes in G and a set of data which are "minimal" with respect to induction. This will be central in the proof of Theorem 5.20, one of our main results. Notice that G acts on B(G) u by simultaneous conjugacy on the pairs and that B(G) u /G is finite. We are going to adapt [23,Corollary 4.6] to the case of the conjugacy action of a group on itself. In [23], Losev described an explicit bijective correspondence between g/G and G-equivalence classes of birationally minimal induction data, i.e. triples (l, ζ, O l ) where l is a Levi subalgebra of g, the orbit O l ∈ N l /L is birationally rigid and ζ ∈ z(l) is such that the induction Ind g l (ζ + O l ) is birational. Our aim is to find an analogue result in the case of the conjugacy action of G on itself.  When G is semisimple simply connected, we will omit the adverb "weakly", i.e. we will say that B(G) is the set of birational induction data of G and that G · (s Ind Now we prove that every conjugacy class is weakly birationally induced in a unique way from a triple of birational induction data, up to conjugacy.
Theorem 4.5. The following map is bijective: In particular, for G semisimple simply connected, every conjugacy class is birationally induced in a unique way from a triple of birational induction data.
where the unipotent classes Ind We can assume that s 1 = s 2 =: s ∈ T and set K := C G (s) • . We have that (7) is equivalent to u2 . Write g = w −1 h for suitable h ∈ M and w ∈ N G (T ) ∩ C G (s), and up to choosing hvh −1 instead of v as a representative, we can assume that g = w −1 ∈ N G (T ) ∩ C G (s). Therefore, we have: Since w acts as an automorphism of K, the induction is birational (Remark 3.13) and O wM2w −1 wu2w −1 is birationally rigid (Remark 3.18), it follows that By Lemma 4.2, the pairs ( The last statement is a consequence of the proof together with Remark 3.9.

Jordan classes and birational sheets
We recall the notions of Jordan classes in a reductive group, introduced in [24] and we collect some results on sheets from [12, §4]. After that, we define birational closures of Jordan classes and birational sheets of a semisimple simply connected group G. The group G acts on D(G) by simultaneous conjugacy on the triples. We associate to any su ∈ G its decomposition data (C G (s)  with O l a rigid nilpotent orbit in l. Definitions and results can be found in [3,4,8,9].
There exist sheets in simple Lie algebras which intersect non-trivially, see [4, §6.6] and [3, §7.4]. Sheets are disjoint in simple Lie algebras of type A, [14]. For g simple of classical type all sheets are smooth, see [18]; if g is simple exceptional there exist singular sheets, see [10] for the list of smooth ones. Similarly, in the case of a simple group G, there exist non-smooth sheets and distinct sheets with non-empty intersection. If G = SL n (C), all sheets are smooth, see [1, §6.3].

Preliminary constructions
Let l be a Levi subalgebra of g and let O l ∈ N l /L. In [23, §4], for any such pair (l, O l ) Losev defines the set Bir(z(l), O l is birational}, by Remark 3.7. In particular, Bir(z(l), O l ) only depends on the pair (l, O l ) and not on the parabolic subgroup chosen for the generalized Springer map. We would like to define a similar object for the group case, but Lemma 3.5 and Remarks 3.9 and 4.4 suggest two distinct approaches.

Birational closures of Jordan classes
In this part, we assume G semisimple simply connected and we apply the results obtained above to define and birational closures of Jordan classes and study their structure.
We continue with other structural results on birational closures of Jordan classes.
Proof. This follows from Definition 5.11 and Proposition 3.16.

Birational sheets
In this section, we still assume G semisimple and simply connected and we prove one of the main result of the work: inspired by [23, §4], we define birational sheets for the conjugation action of G on itself and we prove that they partition G. We start by defining the set: G acts on BB(G) by simultaneous conjugacy and BB(G)/G is finite because D(G)/G is so. Every birational sheet, being irreducible and contained in G (n) for some n ∈ N, is contained in a sheet.
bir is a birational sheet, whereas O G subreg is not so.
All sheets of g contain nilpotent orbits [3, §3.2], but not all birational sheet of g do [23, §4]. Similarly, all sheets of G contain isolated classes, see [11, Proposition 3.1], but we give an example of a birational sheet without this property.
where the first member is J(τ ) while O G s1u and O G s−1v are the two isolated classes of the sheet J(τ ) reg , indeed C G (s 1 ) and C G (s −1 ) are semisimple of type where K ′ ≃ Sp 4 (C) and K ′′ ≃ Sp 2 (C) and decompose Proof. This follows from Example 3.20 and Lemma 5.23 and Theorem 5.20.
Remark 5.25. We claim that Lusztig's strata defined in [25] are disjoint unions of birational sheets. This follows from [11, Proof of Theorem 2.1]: it is proven therein that if J ∈ J (G) lies in a stratum, then J reg lies in that stratum. Since J bir ⊂ J reg , we get that strata are unions of birational closures of Jordan classes. By taking maximal sets with respect to inclusion in this decomposition, we conclude our claim.

Weakly birational sheets
We would like to cast some light on the case in which G is not simply connected. Recall that in this case, birational induction and weakly birational induction are two distinct concepts so that Bir(Z(M ) • s, O M ) as in (9) can be a proper subset of WBir(Z(M ) • s, O M ) as in (10). In §5.2 and §5.3, we treated Jordan classes as the "building blocks" of birational closures and birational sheets. Recall Example 5.5 and retain notation therein: for G = PSL 2 (C), the Jordan class of regular semisimple elements J(τ ) = G · (T \ {ē}) does not consist of all birationally induced conjugacy classes. This implies it is not possible to extend directly constructions and proofs of results in §5.2 and §5.3 by requiring that classes are birationally induced in the sense of Definition 3.8 (a). Nonetheless, the results in §5.1 hold for G not necessarily simply connected, therefore we give the following definition as a possible generalization. Similarly, for any connected reductive group G, it still makes sense to introduce BB(G) ⊂ D(G) as in (11).
Remark 5.29. All results concerning birational sheets proven in §5.3 for G simply connected can be restated for weakly birational sheets in the case of any connected reductive group G; in particular, any connected reductive group is partitioned into its weakly birational sheets. 3 6 Local geometry of birational closures We start this section with definitions and results from algebraic geometry which will be useful for our purposes. Following terminology of [17, §1.7], two pointed varieties (X, x) and (Y, y) are said to be smoothly equivalent if there exist a pointed variety (Z, z) and two smooth maps φ : Z → X and ψ : Z → Y such that φ(z) = x and ψ(z) = y. In this case we write (X, x) ∼ se (Y, y). By [20, Remark 2.1], if dim Y = dim X + d, then (X, x) ∼ se (Y, y) if and only if (X × A d , (x, 0)) and (Y, y) are locally analytically isomorphic. Smooth equivalence is an equivalence relation on pointed varieties and it preserves the properties of being unibranch, normal or smooth. For any algebraic variety X, denote by X an the associated analytic space.
Lemma 6.1. Let X and Y be complex algebraic varieties with dim X = dim Y + d. Let X and Y be unibranch at x ∈ X and at y ∈ Y , respectively. Suppose (X, x) ∼ se (Y, y). Let ψ X : X → X and ψ Y : Y → Y be the normalizations of X and Y , respectively. Letx ∈ X andỹ ∈ Y with ψ X (x) = x and ψ Y (ỹ) = y, respectively. Then ( X,x) ∼ se ( Y ,ỹ).
Proof. By assumption, (X an , x) and (Y an × A d , (y, 0)) are locally isomorphic as analytic pointed spaces. Let X an (resp. Y an ) be the normalization of X an (resp. of Y an ). By [21, §5, Satz 4], we have X an = X an and Y an = Y an . Thus, ( X an ,x) is the analytic normalization of (X, x) and ( Y an × A d , (ỹ, 0)) is the analytic normalization of (Y × A d , (y, 0)). Hence, ( X an ,x) and ( Y an × A d , (ỹ, 0)) are locally analytically isomorphic and this concludes the proof.
Now we make use of the previously introduced instruments to describe the birational closure of a Jordan class around a unipotent element of G connected and reductive. (ii) if v ∈ J wbir , then (J wbir , v) ∼ se (J bir , ν).