The variety of two-step loop-complexes isthe set of pairs of matrices {(X,Y)  |  XY = 0, YX = 0}.Using the construction of Lusztig [4], we show that this varietyis isomorphic to an open subset of a Schubert variety for the loop group of GL_n(C). As an application, we give an explicitBott-Samelson resolution of the loop-complex variety.

1  Affine Schubert varieties

Let F: = C((t)), the field of formal Laurent seriesf(t) = å_{i ³ N}a_j t^j with a_j Î C; and A: = C[[t]], the ring of formal Taylor series.Fix a positive integer n, and define the loop groupG = GL_n(C)^Ù: = GL_n(F),the group of invertible n×n matrices with coefficientsin F.

Let V: = Fⁿ, a vector space over F with a naturalaction of G. Let e₁,¼, e_n denote the standardF-basis of V, and for c Î Z,define e_i+nj: = t^j e_i, so that {e_i}_{i Î Z}is a C-basis of V. With respect to these bases, we may write an element v Î V either as a column vectorv = [a₁,¼,a_n]^T with coefficients a_i Î F,or as an infinite column vector[¼,a_-1,a₀,a₁,a₂,¼]^T with coefficients a_i Î C. (Here ^T means transpose.)We will sometimes think of a matrix g Î G as a listof column vectors: g = (v₁,¼,v_n).

An A-lattice L Ì V is the A-submodule L = Av₁żÅAv_n, where {v₁,¼,v_n} is an F-basis of V.The standard A-lattice is: E: = Span_Aáe₁, ¼e_nñ = Span_Cáe_iñ_{i ³ 1},and we let E_k: = Span_Cáe_iñ_{i ³ k}.

The affine Grassmannian Gr(V) is the spaceof all A-lattices of V. Clearly Gr(V) is ahomogeneous space with respect to the obvious actionof G, and the stabilizer of the standard lattice Eis the maximal parabolic P: = GL_n(A), the subgroup of matrices X with coefficients in A such that detX = a₀+a₁ t+¼ with a₀ ¹ 0.Thus G/P @ Gr(V), and a matrix with column vectors(v₁,¼,v_n) Î G corresponds to the latticewith basis áv₁,¼,v_nñ.

Let a,b be positive integers with a+b = n.The partial affine flag varietyFl(a,b;V) is the space of all pairs of latticesL_· = (L₁,L₂)such that L₁ É L₂ É tL₁ anddim_C(L₁/L₂) = a, dim_C(L₂/tL₁) = b.We write these conditions concisely as:

Now let S_¥ be the group of bijections p:Z®Z,and let t:i® i+n be a shift bijection. Define the disconnected Weyl group W ^~ Ì S_¥ to be the subgroupof bijections which commute with t: that is, W ^~ : = {p Î S_¥  |  pt = tp}.Clearly, any permutation of [1,n] extends in a unique way to an element of W ^~ . Furthermore, any element p Î W ^~ is equivalent to a sequence of integers[p(1),¼,p(n)] such that ^-p^-:i® ^-p(i)^-defines a permutation of [1,n],where ^-m^-: = m mod n. For example, we denotet = [n+1,n+2,¼,2n].

We may embed the W ^~ Ì G.For p Î W ^~ we let p act F-linearly on V by p(e_i): = e_p(i). The correspondingmatrix is the affine permutation matrix (a_ij)with a_^-p^-(i),i = t^c_i, where p(i) = ^-p^-(i)+nc_i.For example, we may identify t = diag(t,¼,t),

Define the simple reflections s₀,¼,s_n-1 in W ^~ by s_i(i) = i+1, s_i(i+1) = i, and s_i(j) = j for j ^not º i, i+1 mod n. We denote s_n: = s₀,and more generally s_i+nj: = s_i.Then s₀,¼,s_n-1 are involutionsgenerating a subgroup W Ì W ^~ and satisfying theCoxeter relations (s_i s_i+1)³ = id for 0 £ i £ n-1, and (s_i s_j)² = id otherwise.We have a semi-directproduct W ^~ = ásñ |×W. Here s:i® i+1 is the shift operator, whichacts on W via the outer automorphism:ss_is^-1 = s_i+1.

By Gaussian elimination, we establish theparabolic Bruhat decomposition of G into double P_a-cosets:G = _-|_-|_{-p Î Wa\W ^~ /Wa} P_apP_a, where we consider each p as an affine permutationmatrix, and W_a = ás₁,¼,s_a-1,s_a+1,¼,s_nñ.We have the corresponding decomposition ofthe affine flag varietyFl(a,b;V) = _-|_-|_{-p Î Wa\W ^~ /Wa} X_pinto Schubert cells X^°_p: = P_a· pE_·,where pE_· = (pE₍₁₎ É pE₍₂₎) is a translation of the standard flag.

The Schubert cells can be defined by Schubert conditions:

For any Schubert variety X_p,we define a certain affine open subset, the opposite cell X¢_p Ì X_p.First we deal with the special case p Î W Ì W ^~ .Let E¢₍₁₎: = Span_Cáe_iñ_{i £ 0}and E¢₍₂₎: = Span_Cáe_iñ_{i £ a}complementary spaces to E₍₁₎,E₍₂₎, and define X¢_p Ì X_p as the set of L_· Î X_p such that L_iÇE¢_(i) = 0 for i = 1,2. For example, E_· Î X¢_p for any p Î W.Next, for the general case of p Î s^k W, we letX¢_p: = { L_· Î X_p  |  L_iÇs^k E¢_(i) = 0 for i = 1,2};so that s^kE_· Î X¢_p.

2  Loop-complexes and Lusztig's isomorphism

For positive integers a £ b, we consider the variety of two-step loop-complexes:

Now, L is a subvariety of the representations of the affine quiver A^Ù₁;a subvariety which is invariant under the natural action of the group GL_a,b(C): = GL_a(C)×GL_b(C), namely(g_a,g_b)·(X,Y): = (g_bXg_a^-1,g_aYg_b^-1).We easily see that L is a finite union of GL_a,b(C)-orbits. In fact, Lhas exactly a+1 open orbits L^°₀,¼,L^°_a, whose closures give the a+1 irreducible components of L:

We define an isomorphism from L to a union of opposite cells of Schubertvarieties in Fl(a,b;V), where V = Fⁿ and n = a+b. Our notation will emphasizethe block decomposition V = F^aÅF^b,as well as:

We easily deduce:

Here

3  Affine permutations for loop-complexes

We wish to identify the image of L_c as the opposite cell of an affine Schubert variety,

To analyze the decomposition of p into simple reflections,we construct the loop wiring diagram of p.That is, the strip below represents a cylinder (identify thetop and bottom edges) with n = a+b dots on either end.For each i we write p(i) = ^-p^-(i)+nj, and we draw a wire connecting the dot ion the right to the dot ^-p^-(i) on the left,but looping upwards (around the cylinder) j times.We will group the wires into five cables correspondingto our blocks i,...,v (on the right)and i¢,¼,v¢ (on the left),so that the cable starting at i represents cnon-crossing wires, etc. As a final simplification,instead of drawing the diagram for p, we instead draw the diagram for t^-1p(a harmless normalization, since t is in the center ofW ^~ ).

Whenever a cable with k wires crossesone with k¢ wires, we have a total of kk¢ wirecrossings. Thus the six cable-crossings of our picturegive a wire-crossing total of:

Now we may write a reduced decomposition for pas follows. For integers i,k, define the affine permutation s_i^[k]: = s_i s_i-2¼s_i-2k+2, which has kmutually commuting factors. Recall our conventions_i+nj: = s_i. For each cable crossing:

4  Bott-Samelson resolution

We can use the above data to give a Bott-Samelsonresolution of singularities for L_c. Although this is clearly far from a minimal resolution,it brings the loop complexes into the framework of Frobenius splittings and other results forBott-Samelson varieties. In particular, thesingularities of L_c are rational, effective line bundles are acyclic, etc.

The construction of the affine Bott-Samelson varietyZ_i corresponding to a reduced word i = (i₁,¼,i_l)is exactly analogous to (and includesas a special case) the construction for GL_n(C).It is best illustrated by an example.

Let us take c = 1, a = 2, b = 3, n = 5 so that each ofthe blocks i,¼,v hassize 1, and our cable diagram in the previoussection is a simple wiring diagram.Then p = s₁ s₃ s₄ s₃ s₀ s₂,and the Bott-Samelson variety is:

We can definea regular, birational map of Z_ionto X_p byforgetting all the spaces except (L₁,L₃). This map is genericallyone-to-one because generically all the spacesare determined by L₁, L₃: that is,L₂ = L₁Ç E₁, L₅ = tL₁+tE₁, etc.To desingularize the opposite cell X¢_p @ L_c, we consider the subset of Z_i whereL₁, L₃ are generic with respect to the opposite standard flag E¢₁, E¢₃.

For general a,b,c, the diagram of inclusionsdefining Z_i is closely related to thedual graph of the wiring diagram of p.

References

[1]

A. Björner and F. Brenti,Affine permutations of type A,El. J. Combin. 3, Foata Festschrift.

[2]

V. Lakshmibai and P. Magyar,Degeneracy schemes, quiver schemes,and Schubert varieties, Internat. Math. Res. Notices12 (1988) 627-640.

[3]

G. Lusztig,Green polynomials and singularities of unipotentclasses, Adv. in Math. 42 (1981) 169-178.

[4]

G. Lusztig,Canonical bases arising from quantizeduniversal enveloping algebras,JAMS 3 (1990) 447-498.