Furthermore, De Concini and Procesi proved that the Picard group Pic(X) is a wonderful compactification of the sublattice Λ×Λ, where Λ is a weight space. In fact, this is the main result of this thesis and appears in Chapter 4, where we find an explicit Frobenius splitting of the compactification of X such that the limit divisors and the closure of U in X are compatible Frobenius splittings. In fact, the last paper (ie [B-P2]) was a big inspiration in our attempt to prove that the beautiful compactification ¯U of the unipotent manifold U is a Frobenius split.
1A fundamental character is the composition of the track chart and the representation associated with the fundamental weight.
Representations
The following Lemma is the criterion on a G-module V for the action of G on P(V) to factorize by the action of Gad. Then the action of G on P(V) factors by the action of Gad if the difference of any two weights of V is in the root lattice. Assume that any difference of two weights of the G-module V in the root lattice is X∗(Tad).
Since φi is B × B-eigenvector, we have that the closure of Bsiw˙ oB inG is contained in the zero subset VG(φi).
Filtrations and Tilting Modules
Since the fundamental weights form a basis for the weight space Λ, we conclude that only one of theχij is non-zero, i.e. Therefore VG(φ) is the closure in G of Bsiw˙ oB by [Spr] Proposition 1.8.2 since both sets are closed, irreducible and of the same dimension (i.e. codimension 1). For the construction of the wonderful condensation X of Gad as well as of the properties of X, we follow [B-K] chapter 6.
The next section will show that X satisfies the rest of the properties of the wonderful compression, e.g.
Properties of X
So we only need to prove the last statement for Xo, namely that X is the union of G×G-. If Y is the unique closed path of a G-manifold X and if V is an open subset of X satisfying V ∩Y 6=∅, then X =S. Thus the closure of the G×G circuit of [h]∈P(Endk(M)) is isomorphic to the variety X constructed above.
Therefore, the closure of the G×G-orbit of the element ([h1],[h2]) maps isomorphically to the closure of the G×G-orbit [h], which is of variety X.
The Picard group of X
The irreducible components of X\Xo are the prime divisors Bs˙iB− where si is the simple reflection in W associated with αi. This implies that the closure in C of Gad∩C is the empty set since C is irreducible. But then C is one of the divisors Xi since it has codimension 1 in X by [Ha2] Proposition II.3.1.
To complete the proof, we need to prove that Bs˙iB− are linearly independent in Cl(X). But then the constraint if|Xo is a non-constant invertible function of the affine spaceXo which gives a contradiction. The restriction of L to the unique closed orbit Gad×Gad Y 'G/B×G/B is also a G×G-linearized line group in Y .
Let τi denote the unique (up to a scalar) section of LX(Di) such that the zero subset of τi is Di. Note that the classes of Di form a basis of Cl(X) since Di = (1,w˙o)BsiB− is linearly equivalent to BsiB−. Suppose further that σi denotes the unique (up to a scalar) section of the line beam LX(Xi), so that the zero subset of σi is Xi.
We also prove that the central isogeny π :G→Gad maps U bijectively onto the unipotent variety Uad of the adjoint linear algebraic group Gad. A subset U of all unipotent elements in a connected reductive linear algebra group G is a closed irreducible subvariety of G dimensiondim(G)−`.
Properties of the Unipotent Varieties
The fundamental character is then defined to be the composition of homomorphism of algebraic groups and the trace map, i.e. an element g in G is regular if the dimension of centralizerCG(g) is equal to the rank of G. Here we have predetermined an order of the positive roots as in the proof of Lemma 3.1.2 on page 26.
Therefore, u0 ∈Ru(Pj), the unipotent radical of the minimal parabolic subgroup Pj = B∪Bs˙jB cf. Note that in the first part of the proof we proved that dim(CG(x))≥. It further induces the bijection of the unipotent conjugacy classes G and the unipotent conjugacy classes Gad.
We must take squares by the description of the coordinate ring k[Gad] in [Spr] Exercise 2.1.5(3). We use a result due to Strickland ([Str]) which allows us to construct Frobenius splittings of X by expanding a Frobenius splitting of the unique closed orbitY =G/B×G/B. First, we introduce the general concept of Frobenius distributions in section 4.1 which summarizes some of the well-developed theory (see [M-R], [RR] and [R]).
To obtain control over the Frobenius splittings of X, we find an explicit measure for Frobenius splittings of Y and further, aG×G equivariant map f :StSt→H0(X,LX((p−1)ρ) ) so that the composition with the constraint map to G/B×G/Bis is non-zero. Having established the notation, we can give a local description of the evaluation map HomOZ(F∗OZ,OZ) → OZ(Z) given by φ 7→ φ(1).
Frobenius splittings of Y and of X
Before answering that question, we need an observation: Since St is self-dual ([Jan] §II.3.18 and Corollary II.2.5), we can choose an isomorphism γ :St →St∗ which then yields a unique G -equivariant bilinear formχ:St St. →k given by χ(u⊗v) = γ(u)(v). Now τ is also G-equivariant under the canonical G-structures of OG/B and F∗OG/B and by Theorem 4.1.5 on page 38. Since the G-equivariant for χ : St⊗St → k is unique (by Frobenius reciprocity), the composition τ ◦m is equal to χ up to a scalar z ∈ k.
We can see the module StStas aG via the action of the diagonal ∆(G) of G, which we then denote St⊗St. Since by Frobenius reciprocity ([Jan] Proposition I.3.4b) St⊗St contains a unique (up to scalar) G-invariant element. Note that the ∆(G)-invariant element v ∈ StSt corresponds to the identity endomorphism in End(St)'St∗⊗St'St⊗St.
In the proof of Theorem 2.3.2 on page 21, we have that the global section τiofLX(Di)'LX(ωi) is a B×B eigenvector of weight (ωi,−woωi). When we first dualize and then use Frobenius reciprocity ([Jan] Theorem I.3.4b) on this map, we get the G×G homomorphism. This provides a good check on F-splitting sections of X constructed as in Theorem 4.2.1 on page 40 because resY ◦f is the identity.
Hence the G-invariant bilinear form χ:St⊗St →k is given by (ab)⊗(xy)7→ay−bx since. Note that atf is equal to the map in Lemma 4.3.2 on page 45, which explains why we get this result.
Frobenius splitting of U ¯
By identifying St⊗St ' St∗ ×St ' End(St) where the first map is γ ⊗id and observing that the identity map is the ∆(G)-invariant element in End(St), we find that v = Pd . Hence, we seek global cuts si of X such that resGad ◦ π are the fundamental characters χi on G due to Proposition 3.2.3 on page 29. Since g is a G⊗G-equivariant map and sincev ∈Stis the unique ( up) to scalar) ∆(G)-invariant element we only need to show that g(t) is non-zero when restricted to Y .
When writing t in a basis consisting of weight vectors, the coefficient of the highest weight vector of Z is non-zero according to the above observation. Since Gis is simply connected, the Picard group of Gis trivially implies that up to a scalar level we have the following isomorphism. We have now reached our main result, namely that the closure in X of the unipotent manifold Uad or Gad is compatible F-split in X when G=Slq.
Furthermore, every irreducible component of ∩`i=1VX(gi(ti)) is compatible F-split according to Proposition 4.1.6 on page 39. Since the zero subset of σi is the irreducible divisor Xi per definition, Xi is compatible F-split by Proposition 4.1.6 on page 39. Its closure in X satisfies the same, and so ¯U is an irreducible component of the intersection ∩` i=1VX(gi(ti)) (has the highest possible codimension of Krull's Hauptidealsatz).
It is worth noting that the manifold ∩`i=1VX(gi(ti)) is F split in all positive features. The assumption that χi(e) = 0 in k for all i is quite restrictive in that it is only satisfied in the following case as shown in appendix A.
U ¯ is B-Canonical split
So if χi(e) = 0 in k for all i, the closure is U¯ of the unipotent variant of Gad B-canonical splitting. We find that the main result of the previous section implies that ¯U is a local full intersection and normal. We also have a partial result on the Picard group of ¯U where ¯U is assumed to be a local complete intersection.
Then if the schematic intersection contains Z :=Y∩Dis reduced, then every irreducible component of Z contains a smooth point of Y. Note that every irreducible component of Z contains a smooth point in Z, since the smooth points are a dense subset of a reduced schedule. To prove the statement, note that ¯U is actually an irreducible part of V, since it is irreducible, closed, and codimension (inX).
But then C ⊆V ∩Xj is an irreducible component since it is a maximal closed and irreducible subset and thus has codimension `+ 1 according to Lemma 5.1.1 on page 53 which contradicts codim(C, X). Now the previous Lemma 5.1.2 gives that every irreducible component of the intersections ¯U∩Xi contains smooth points of ¯U. Let ¯Using be the singular place of ¯U and let C denote an irreducible component of ¯Using.
Recall that the regular elements Ureg and U form an open G-orbit (in U) by Proposition 3.2.7 on page 31. If C∩Gad =∅, then C is a closed and irreducible subset of Xi∩U¯ and is therefore contained in irreducible components C0 of Xi ∩U¯.
The Picard group of U ¯
Using [Ha1] of Exercise II.6.8(a), we obtain from the diagram in (5.1) the following commutative diagram of homomorphisms. Note that the fundamental character evaluated on the identity elements of G is nothing more than the dimension of the representation corresponding to the fundamental weight, ωi, which means that it may be more natural to switch the point of view to the simply connected group G v instead of the connected group Gad.
But for a simply connected group G there is no canonical compactification such as the wonderful compactification of Gad. More generally, question 1 can be reformulated as "Are Steinberg fibers F-split in any reductive embedding1 and any positive characteristic?". If we can give an explicit description of the Picard group ¯U, then we can probably prove some vanishing results.
One can also take a closer look at the good filtration that bundles the global parts of a spacious line bundle on ¯Uadmit. It can be proved that the setZ0 considered in the proof of Theorem 3.1.1 on page 25 is actually a desingularization of U. This problem is related to finding an equivariant desingularization of the large Schubert variety ¯Bad; see [B-P2].
G is a connected, semisimple, simply connected linear algebraic group over k with a surjective morphism of algebraic groups π:G→ Gad and the kernel of π is central. X∗(Tad) The characters of Tad is the root lattice, the root-spanned sublattice of Λ over Z. X = X \Gad; the Gad boundary which is a union of the stable Gad×Gad-stable quotient with normal transitions.
Yes Open affine subgroup of P(Endk(M)) such that when the element is written in the basis defined in Remark 2.1.4, then the coefficient of hλ =m∗λ ⊗mλ is equal to 1.
