Affine Springer Fibers and the Affine Matrix Ball Construction for Rectangular Type Nilpotents
aa r X i v : . [ m a t h . R T ] A ug AFFINE SPRINGER FIBERS AND THE AFFINE MATRIX BALLCONSTRUCTION FOR RECTANGULAR TYPE NILPOTENTS
PABLO BOIXEDA ALVAREZ, LI YING, AND GUANGYI YUE
Abstract.
In this paper, we study the affine Springer fiber F l N in type A for rectangu-lar type semisimple nil-element N and calculate the relative position between irreduciblecomponents. In particular, we use the affine matrix ball construction to show the relativeposition map is compatible with the Kazhdan-Lusztig cell structure, generalizing the workof Steinberg and van Leeuwen. Contents
1. Introduction 12. Combinatorial preliminaries 33. Structure of two-sided Kazhdan-Lusztig cells of rectangular type 94. Affine Springer fibers and their geometry 165. Components of affine Springer fiber of rectangular type 186. The case of n = 2 w λ Introduction
In the paper [13], Kazhdan and Lusztig laid a foundation for studying representations ofHecke algebras, and in particular they introduced the notion of (two-sided, left, right) cellsfor Coxeter groups. In type A , the Kazhdan-Lusztig cell structure of the symmetric groupcorresponds to the well known Robinson-Schensted correspondence, which is a bijectionbetween the symmetric group S n and pairs of standard Young tableau of the same shape λ ⊢ n : w ∈ S n ( insertion tableau P, recording tableau Q ) . Namely,(1) two permutations are in the same two-sided cell iff they have the same associatedpartition λ ;(2) two permutations are in the same right cell iff they have the same insertion tableau P ;(3) two permutations are in the same left cell iff they have the same recording tableau Q .Robinson-Schensted correspondence are realized by many equivalent combinatorial algo-rithms, for example the row-insertion algorithm and the matrix ball construction [7,24]. This ombinatorial correspondence appears in the study of (finite) Springer fibers. Given a nilpo-tent N of type λ , Spaltenstein [21] labeled the irreducible components of the Springer fiberof N by standard Young tableau of shape λ . Later on, Steinberg [22] showed the relativeposition between two components labeled by tableaux P and Q respectively are exactly thepermutation corresponding to ( P, Q ) under the Robinson-Schensted algorithm. This resultis compatible with the cell structure since the image of the relative position map is exactly aright (resp. left) cell if we fix the first (resp. second) component. These nice interpretationsare further extended by van Leeuwen in [23]. And the natural question is to find an analoguein the affine setting.On the combinatorial side, the Robinson-Schensted correspondence is generalized by Shi[19] to the affine symmetric group f S n , giving a parametrization of the left cells by tabloids.The shape of these tabloids determines the two-sided cell. Later Honeywill [12] added thethird piece of data, weights, to make it a bijection: w ∈ f S n ( insertion tabloid P, recording tabloid Q, dominant weight ρ ) . Both Shi and Honeywill’s algorithms are very involved and Chmutov, Pylyavskyy, Yudov-ina [5] generalized the matrix ball construction given by Viennot to give a simpler and moreintuitive realization. This generalized algorithm, named the affine matrix ball construction,has a variety of nice applications. In particular, it is used to understood the structure ofbi-directed edges in the Kazhdan-Lusztig cells in affine type A in [4]. Most importantly,fibers of the inverse map of affine matrix ball construction possess a Weyl group symmetry.The relative position map in the affine setting, though not injective, is proven to have thesame property, which is our motivation to establish a bijection similar to Steinberg’s.On the geometry side, the affine Springer fibers appearing in this paper have been studiedbefore. In particular the geometry of these are studied in [9]. The case for type (1 n ) hasbeen studied in further depth. In particular the cohomology has been studied by worksof Goresky, Kottwitz, Macpherson [8], Hikita [11] and Kivinen [15]. This affine Springerfiber is also related with the representation theory of small quantum groups as proven inupcoming work of Bezrukavnikov, McBreen and upcoming jont work of the first author withBezrukavnikov, Shan and Vasserot [1].In [18], Lusztig introduced the partitions of the extended affine Weyl group f W into S -cellsand ˜ S -cells, both parametrized by the conjugacy classes in the finte Weyl group W , andconjectured that the image of the relative position map from pairs of irreducible componentsof the affine Springer fiber to the extended affine Weyl group Irr( F l N ) × Irr( F l N ) → f W is exactly the S -cell of type γ where N is a regular semi-simple nil-element of type γ . Lusztig’sconjecture is proved in the recent work of Finkelberg, Kazhdan and Varshavsky [6] in generaltype. They use the affine Springer resolution and families of Springer fibers, but here wefocus on a single affine Springer fiber. Also Lawton [16] showed that in type ˜ A , the two-sidedcells and the ˜ S -cells coincide.It follows directly from [6] and results in this paper that the S -cells and the two-sidedKazhdan-Lusztig cells agree in type ˜ A and for rectangular type, which also coincides with ˜ S -cells by [16]. The general relationship between S -cells and two-sided cells is still unknown,and we leave this for future investigation. n this paper, we focus on type A and N is of rectangular type ( l m ) , and study therelationship between relative position and the two-sided, left and right cells instead of S -cells, giving an affine generalization of Spaltenstein, Steinberg and van Leeuven’s resultsin the finite case. The special column type (1 n ) is presented in [2, 3, 25]. Namely, we useaffine matrix ball contruction to establish a bijection between pairs of irreducible componentsmodulo common translations with Ω ( l m ) , which are triples ( P, Q, ρ ) of rectangular-type and ρ is not necessarily dominant. This is given in the following commutative diagram: Irr( F l N ) / Λ Irr( F l N ) × Λ Irr( F l N ) f S n T ( l m ) Ω ( l m ) θ pr i r Θ pr i Ψ where T ( l m ) is the collection of tabloids of shape ( l m ) , Ψ is the inverse of the affine matrixball construction, and pr i , i = 1 , , are the projections onto the first and second componentrespectively. It follows that the image of the relative position map r is the two-sided cellof type ( l m ) . Moreover, r ( pr − ( C )) (resp. r ( pr − ( C )) ) is a right (resp. left) cell for any C ∈ Irr( F l N ) / Λ , similar to the finite scenario.The rest of the paper is organized as follows. In Section 2, we review the affine matrixball construction and the related combinatorics about the affine symmetric group. And inSection 3 we study the explicit structure of two-sided cell of rectangular type. Section 4is a review of basics on affine Springer fibers and in Section 5 we study the geometry ofthe irreducible components of F l N when N is of rectangular type and calculate the relativeposition between any two irreducible components. Section 6 deals with the case of n = 2 explicitly. The proof of the main theorem is presented in Section 7. In the appendix, wegive diagrams of left Knuth classes containing w λ when λ = (2 , , (3 , and (2 , , . Acknowledgements.
The authors would like to thank Roman Bezrukavnikov for sug-gesting this problem and continuous discussions throughout the process. Also, the authorsare grateful to Zhiwei Yun for many useful discussions. The first author also wants to thankDongkwan Kim and Pavlo Pylyavskyy for a useful early discussion.2.
Combinatorial preliminaries
For the entire paper, we fix a positive integer n . Denote [ a, b ] = [ a, a + 1 , . . . , b ] for any a, b ∈ Z , a < b and [ a ] = [1 , a ] for a ∈ Z > . For any i ∈ Z , let i be the residue class i + n Z ,and denote [ n ] = { , . . . , n } . Affine Symmetric Group.
Let S n be the symmetric group on n letters, which isthe Weyl group of type A n − . The extended affine symmetric group S n is the collection ofall bijections w : Z → Z satisfying w ( i + n ) = w ( i ) + n for all i ∈ Z . And we call theelements in S n to be extended affine permutations . Let f S n ⊂ S n be the affine symmetricgroup consisting of all w ∈ S n satisfying P ni =1 w ( i ) = n ( n +1)2 , and elements inside f S n arecalled affine permutations . The (extended) affine symmetric group is exactly the (extended)affine Weyl group of type ] A n − . Since (extended) affine permutations are determined by its values on [ n ] , we use the windownotation [ w (1) , . . . , w ( n )] to represent w . We denote w = h w (1) , . . . , w ( n ) i ∈ S n . he affine symmetric group f S n is the Coxeter group generated by simple reflections s , . . . , s n − , s = s n (we take the indices i in [ n ] without ambiguity) under Coxeter rela-tions, where s i can be viewed as a permutation on Z such that s i ( x ) = x + 1 , x ≡ i (mod n ) ,x − , x ≡ i + 1 (mod n ) ,x, else.And S n = Ω ⋉ f S n where Ω is the infinite cyclic group generated by s = [2 , , . . . , n + 1] . The rotation map φ ( w ) = sws − is an automorphism of f S n sending s i to s i +1 for i ∈ [ n ] , whichcorresponds to the rotation of the Dynkin diagram of type ] A n − .There are two well-known formulas for computing the length of an affine permutation, thefirst one is given by Shi [19]: Lemma 2.1.
For w ∈ f S n , we have ℓ ( w ) = X ≤ i
We now follow[4,5] and identify affine permutations with its matrix ball configuration. In detail, for w ∈ f S n ,we draw a Z × Z matrix with row labels increasing southwards and column labels increasingeastwards. If w ( i ) = j , we draw a ball in the ( i, j ) -position of the matrix and will be namedalso by ( i, j ) without ambiguity. And we denote B w = { ( i, w ( i )) | i ∈ Z } which is thecollection of the balls of w . The periodicity of w implies that ( i, j ) ∈ B w iff ( i + n, j + n ) ∈B w . We say ( i + kn, j + kn ) for k ∈ Z are the ( n, n ) - translates of ( i, j ) . For two balls ( i, j ) , ( k, l ) ∈ B w , we define the southeast (partial) ordering ≤ SE by ( i, j ) ≤ SE ( k, l ) iff i ≥ k and j ≥ l , i.e. ( i, j ) is southeast of ( k, l ) . Other relations using compass directions can bedefined similarly, and are also partial orders on Z × Z .A partition λ of size n ∈ N is a finite tuple of weakly decreasing positive integers λ =( λ , ..., λ k ) with sum n . Denote ℓ ( λ ) = k to be the number of nonzero parts of λ . The Youngdiagram of a given partition λ is a left-justified collection of boxes with the first row having λ boxes, second row having λ boxes and so on. And we denote λ T to be the transpose of λ . For a given partition λ of size n , a tabloid of shape λ is an equivalence class of bijectivefillings of the Young diagram of λ with [ n ] , such that two fillings are equivalent if one isobtained from the other by permuting the entries of each row.We denote the collection of all tabloids of shape λ to be T ( λ ) . And let T λ ∈ T ( λ ) be thetabloid with in the first row, in the second row,..., λ T1 in the last row, λ T1 + 1 in the firstrow and so on. There is a natural left action of S n on T ( λ ) and for X ∈ T ( λ ) , X + k isdefined to be the tabloid adding k to each entry in X .For any tabloid X ∈ T ( λ ) , i ∈ [ λ T1 ] , let X i ⊂ [ n ] be the i -th row of X . Denote X i = (cid:8) X i, , X i, , . . . , X i,λ i (cid:9) such that X i, , . . . , X i,λ i ∈ [ n ] and X i, < . . . < X i,λ i . Moreover,throughout the paper, we always extend the column indices as: X i,j + kλ i = X i,j + kn for i ∈ [ λ T1 ] and k ∈ Z . xample 2.2. For λ = (3 , , , , T λ and a tabloid X in T ( λ ) are the following: T λ = 1 8 52 67 34 , X = 7 1 42 56 38 . Then X = (cid:8) , , (cid:9) , X = (cid:8) , (cid:9) ⊂ (cid:2) (cid:3) and X , = 3 , X , = 6 , X , = 11 , X , = 16 . Definition 2.3.
Given an affine permutation w , a subset C ⊂ B w is called a stream if it isinvariant under ( n, n ) -translations and forms a chain under the southeast partial ordering ≤ SE . The number of distinct ( n, n ) -translation classes of a stream C is called the density of C . A subset C ′ ⊂ B w is called a anti-stream if it forms a chain under the southwest partialordering ≤ SW . The number of entries in an anti-stream C ′ (which is always finite) is calledthe density of C ′ .From Lusztig we could associate λ ( w ) = ( d , d − d , d − d , . . . ) to any affine permutation w where d i is the maximal one among the sums of densities of i disjoint streams in B w . [10,Theorem 1.5] guarantees that λ ( w ) is a partition and is called the partition associated to w .Moreover, ( d ′ , d ′ − d ′ , . . . ) , where d ′ i is the maximal one among the sums of densities of i disjoint anti-streams in B w , is the partition λ ( w ) T .To determine the cell structure, we recall the affine matrix ball construction Φ from[5], which is a nice generalization of Viennot’s geometric construction [24] of the classicalRobinson-Schensted correspondence. Φ : f S n → Ω = G λ ⊢ n ( P, Q, ρ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P, Q ∈ T ( λ ) , ρ ∈ Z ℓ ( λ ) , ℓ ( λ ) X i =1 ρ i = 0 w ( P ( w ) , Q ( w ) , ρ ( w )) . We call P ( w ) , Q ( w ) , ρ ( w ) to be the insertion tabloid , recording tabloid and weight of w respectively. And the common shape of P ( w ) and Q ( w ) is the associated partition λ ( w ) .These satistics give the structure the Kazhdan-Lusztig cells: Theorem 2.4 ( [5, 17, 19, 20]) . (1) Two affine permutations are in the same two-sidedcell iff they have the same associated partition. (2)
Two affine permutations are in the same left (resp. right) cell iff they have the samerecording (resp. insertion) tabloid.
We denote C λ to be the two-sided cell containing affine permutations with associatedpatition λ , and L X (resp. R X ) to be the left (resp. right) cell containing affine permutationswith recording (resp. insertion) tabloid X . In particular, the longest element w λ = [ λ T1 , λ T1 − , . . . , , λ T1 + λ T2 , λ T1 + λ T2 − , . . . , λ T1 + 1 , . . . , n, n − . . . , n − λ T λ + 1] in S λ T1 × S λ T2 × . . . × S λ T λ ֒ → f S n is contained in L T λ ∩ R T λ .We refer the readers to [5] for the details of the algorithm of computing Φ and only statethe formula in a very special case which we will use later. Lemma 2.5.
Suppose n = ml for some m, l ∈ Z > and w ∈ f S n satisfies the following twoconditions: w ( i ) < w ( i + m ) < . . . < w ( i + m ( l − < w ( i ) + n for i ∈ [ m ] ; (2) w (1 + ( j − m ) > w (2 + ( j − m ) > . . . > w ( jm ) for j ∈ [ l ] .Then λ ( w ) = ( l m ) and Φ( w ) = ( P, T λ , ρ ) where P i = n w ( m + 1 − i ) , w (2 m + 1 − i ) , . . . , w ( n + 1 − i ) o ,ρ i = l X j =1 (cid:24) w ( jm + 1 − i ) n (cid:25) − l, for i ∈ [ m ] . Now we describe the Weyl group symmetry of the fiber of the inverse of Φ . Definition 2.6 ( [4]) . Suppose λ i = λ i +1 . For X ∈ T ( λ ) , the local charge lch i ( X ) at row i isdefined to be the smallest integer satisfying X i +1 ,j +lch i ( X ) ≥ X i,j for all j ∈ Z . And the chargematching at row i is the map { X i,j | j ∈ Z } → { X i +1 ,j | j ∈ Z } via X i,j X i +1 ,j +lch i ( X ) . If λ i > λ i +1 , we define lch i ( X ) = 0 . Definition 2.7.
Given X ∈ T ( λ ) , the symmetrized offset constant s ( X ) ∈ Z ℓ ( λ ) of X isdefined as: s i ( X ) = i − X j = i ′ lch j ( X ) , where i ′ is the first row in λ with length λ i . The charge of X is defined to be: charge( X ) = ℓ ( λ ) − X i =1 i · lch i ( X ) . Definition 2.8.
The weight ρ in the triple ( P, Q, ρ ) is called dominant , if ρ − s ( P ) + s ( Q ) is increasing segmentwise according to the part sizes of λ , i.e., for each i , either λ i > λ i +1 ,or λ i = λ i +1 and ( ρ − s ( P ) + s ( Q )) i < ( ρ − s ( P ) + s ( Q )) i +1 . And we define Ω dom := G λ ⊢ n { ( P, Q, ρ ) ∈ Ω | ρ is dominant in ( P, Q, ρ ) } . The dominant representative ρ ′ of ρ in the triple ( P, Q, ρ ) can be computed by ρ ′ = ( ρ − s ( P ) + s ( Q )) dom + s ( P ) − s ( Q ) , where ( ρ − s ( P ) + s ( Q )) dom is the segmentwise increasing rearrangement of ρ − s ( P ) + s ( Q ) according to part sizes of λ. In fact, Φ is a bijection between f S n and Ω dom , and its inverse can be extended to Ψ : Ω → f S n . We refer readers to [5] for details of Ψ and only point out the following crucial result: Theorem 2.9 ( [5]) . For any w ∈ f S n , Ψ(Φ( w )) = w, and for any triple ( P, Q, ρ ) ∈ Ω , wehave Φ(Ψ(
P, Q, ρ )) = (
P, Q, ρ ′ ) , where ρ ′ is the dominant representative of ρ. The inverse of permutations behaves nicely under affine matrix ball construction:
Proposition 2.10.
For w ∈ f S n , Φ( w − ) = ( Q ( w ) , P ( w ) , ( − ρ ( w )) ′ ) where ( − ρ ( w )) ′ is thedominant representative of − ρ ( w ) in the fiber (of Ψ ). e define e ρ ( w ) = ρ ( w ) − s ( P ( w )) + s ( Q ( w )) , to be the centralized weight of w , which is a segmentwise increasing vector according to thepart sizes of λ , and the above proposition is equivalent to saying e ρ ( w ) = − e ρ ( w − ) s. rev , where e ρ ( w − ) s. rev is the segmentwise reverse of the vector e ρ ( w − ) .We end this subsection with the following result on how rotation interacts with affinematrix ball construction. Lemma 2.11.
For any w ∈ f S n and k ∈ [0 , n − , there is Φ (cid:0) φ k ( w ) (cid:1) = (cid:0) P ( w ) + k, Q ( w ) + k, ρ ( w ) + δ k ( P ( w )) − δ k ( Q ( w )) (cid:1) where δ ki ( X ) = λ i ( w ) X j =1 [ n − k +1 ,n ] ( X i,j ) , i ∈ [ ℓ ( λ ( w ))] . Proof.
The matrix balls of φ k ( w ) come from that of w by shifting southwestwards by ( k, k ) .The relative positions of the matrix balls do not change, so do the numberings at each step of Φ in [5]. Therefore the coordinate of the back corner posts at each step of the algorithm willshift southeastwards by ( k, k ) as well. Hence P ( φ k ( w )) = P ( w ) + k , Q ( φ k ( w )) = Q ( w ) + k ,and ρ i ( φ k ( w )) = ρ i ( w ) + λ i ( w ) X j =1 , if Q i,j ( w ) ∈ [ n − k ] , P i,j ( w ) ∈ [ n − k + 1 , n ] ; − , if P i,j ( w ) ∈ [ n − k ] , Q i,j ( w ) ∈ [ n − k + 1 , n ] ;0 , otherwise. (cid:3) Knuth Equivalence Classes.
First we define decent sets for both permutations andtabloids.
Definition 2.12.
Given an affine permutation w , we define its right descent set R ( w ) and left descent set L ( w ) as: R ( w ) = (cid:8) i ∈ [ n ] | w ( i ) > w ( i + 1) (cid:9) ,L ( w ) = (cid:8) i ∈ [ n ] | w − ( i ) > w − ( i + 1) (cid:9) . Given a tabloid X , the τ - invariant of X is defined as: τ ( X ) = (cid:8) i ∈ [ n ] | i lies in a strictly higher row than i + 1 in X (cid:9) . Decent sets interact nicely with affine matrix ball construction:
Proposition 2.13 ( [4]) . For w ∈ f S n , L ( w ) = τ ( P ( w )) and R ( w ) = τ ( Q ( w )) . Now we are able to define Knuth moves.
Definition 2.14.
Two affine permutations w and ws i (resp. s i w ) are connected by a right(resp. left) Knuth move at position i if R ( w ) and R ( ws i ) (resp. L ( w ) and L ( s i w ) ) areincomparable under the containment partial ordering. We therefore have an equivalenceclass named right (resp. left) Knuth class generated by right (resp. left) Knuth moves ∼ RKC ws i (resp. w ∼ LKC ws i ), and we denote RKC w (resp. LKC w ) to be the right (resp.left) Knuth class containing w .Two tabloids X and X ′ are connected by a Knuth move if for some i , X is obtainedfrom X ′ by interchanging i and i + 1 and τ ( X ) and τ ( X ′ ) are incomparable. We call anequivalence class generated by Knuth moves, a Knuth class .The following remarkable theorem describes how the image of an affine permutations underaffine matrix ball construction behaves after a Knuth move (we state the left Knuth versionfor convenience of the calculations later).
Theorem 2.15 ( [4]) . Suppose w ∼ LKC s k w , then: (1) Q ( w ) = Q ( s k w ) ; (2) P ( s k w ) differs from P ( w ) by a Knuth move exchanging i and i + 1 for some i ∈{ k − , k, k + 1 } ; (3) ρ ( s k w ) = ρ ( w ) if i = n , otherwise ρ ( s k w ) differs by ρ ( w ) by subtracting 1 from row k ′ and adding 1 to row k , where i = n lies in row k in P ( w ) and i + 1 = 1 lies inrow k ′ in P ( w ) . Theorem 2.15 tells a right (resp. left) cell is a disjoint union of right (resp. left) Knuthclasses. And [4] gives a complete characterization of Knuth classes by specifying what Q and ρ could be like in each Knuth class. Denote d λ = gcd( λ T1 , λ T2 , . . . ) . Theorem 2.16 ( [4], Theorem 8.6) . Let
X, X ′ ∈ T ( λ ) , then X and X ′ are in the sameKnuth class iff charge( X ) ≡ charge( X ′ ) (mod d λ ) . Definition 2.17.
Given w ∈ f S n , we define the monodromy group G Rw based at w to be G Rw = { ρ ( w ′ ) − ρ ( w ) | w ′ ∈ RKC w , Q ( w ′ ) = Q ( w ) } . Theorem 2.18 ( [4], Theorem 7.28) . For any w with associated partition λ , we have G Rw = ( k X i =1 a i m i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a i ∈ Z , k X i =1 a i m i = 0 ) where m > m > . . . > m k are distinct part sizes of λ T and m i ∈ Z ℓ ( λ ) with 1’s in the first m i rows and 0 after on. In particular, when λ is a rectangle, the monodromy group is trivial. Theorem 2.18 indicates the right (resp. left) Knuth classes of rectangular type permuta-tions are finite, which is the foundation of our study.
Lemma 2.19.
Suppose w ∈ f S n satisfies w ( i ) < w ( i + m ) < . . . < w ( i + m ( l − < w ( i ) + n, ∀ i ∈ [ m ] and u ∈ ( S m ) l is the unique permutation such that wu (1 + ( j − m ) < wu (2 + ( j − m ) < . . . < wu ( jm ) , ∀ j ∈ [ l ] . Then wu ( i ) < wu ( i + m ) < . . . < wu ( i + m ( l − < wu ( i ) + n, ∀ i ∈ [ m ] . roof. Fix any j ∈ [0 , l − , and denote a i = w ( i + jm ) , and b i = w ( i + ( j + 1) m ) for i ∈ [ m ] . Let a σ (1) < a σ (2) < . . . < a σ ( m ) and b η (1) < b η (2) < . . . < b η ( m ) for some σ, η ∈ S m .It suffices to show that a σ ( i ) < b η ( i ) for all i ∈ [ m ] and this reduces to showing there are atmost i − elements in { b , . . . , b m } that are smaller than a σ ( i ) . This is true because we know b σ ( i ) , b σ ( i +1) , . . . , b σ ( m ) are larger than a σ ( i ) . (cid:3) Structure of two-sided Kazhdan-Lusztig cells of rectangular type
In this section we restrict ourselves to n ≥ and study the structure of left (resp. right)Kazhdan-Lusztig cells and left (resp. right) Knuth classes inside the two sided cell C λ when λ = ( l m ) ( n = lm ) is a rectangle. The case of n = 2 is addressed separately in Section 6. Lemma 3.1.
For any partition λ , let d λ = gcd( λ T1 , λ T2 , . . . ) . Then (1) charge( φ ( X )) ≡ charge( X ) − d λ ) for any X ∈ T ( λ ) . (2) charge( X ) (mod d λ ) for X ∈ T ( λ ) is equi-distrubuted, i.e. { X ∈ T ( λ ) | charge( X ) ≡ r (mod d λ ) } is independent of r ∈ (cid:2) d λ (cid:3) .Proof. It suffices to prove the first result. Let λ = ( a b , a b , . . . ) where a > a > . . . . Weassume all b i > , otherwise d λ = 1 and the claim is trivial. Suppose the row in X containing n has length a i . If n does not lie in row b + . . . + b j , then charge( φ ( X )) = charge( X ) − ;otherwise charge( φ ( X )) = charge( X ) − b + . . . + b j . Hence the claim follows from d λ | b + . . . + b j . (cid:3) For the rest of the section, we restrict ourselves to λ = ( l m ) . From Theorem 2.16 andTheorem 2.18, we know there is a bijection: LKC w λ → { X ∈ T ( λ ) | m | charge( X ) } by w P ( w ) . Hence from Lemma 3.1, we know w λ = T ( λ ) /m = n ! m ( l !) m . In order tounderstand the Knuth class containing w λ better, we need to construct explicitly the inverseof the above bijection.We define the following set of affine permutations and study its combinatorial properties: Definition 3.2.
Define the fundamental box F to be the set of affine permutations w satis-fying the following three sets of conditions:(1) w − ( i ) < w − ( i + m ) < . . . < w − ( i + m ( l − < w − ( i ) + n for i ∈ [ m ] ;(2) w − (1 + ( j − m ) < w − (2 + ( j − m ) < . . . < w − ( jm ) for j ∈ [ l ] ;(3) lch i ( P ( w − w λ )) = Diff i ( w − ) for all i ∈ [ m − where Diff i ( w − ) = l − X j =0 (cid:18)(cid:24) w − ( i + 1 + jm ) n (cid:25) − (cid:24) w − ( i + jm ) n (cid:25)(cid:19) . (1)The name fundamental box comes from the equivalent definition using the notion of al-coves, which we explain in Section 5. We can strengthen the second condition in Definition3.2 to the following: Lemma 3.3.
Given w ∈ F − , then we have w ( i + 1 + jm ) − w ( i + jm ) ≤ n − l for i ∈ [ m − , j ∈ [0 , l − . roof. Suppose on the contrary that w ( i + 1 + jm ) − w ( i + jm ) ≥ n − l + 1 for some i ∈ [ m − , j ∈ [0 , l − . Let a x = P i,x ( ww λ ) , b x = P i +1 ,x ( ww λ ) for x ∈ Z . Then w ( i + jm ) = a r and w ( i + 1 + jm ) = b r + γ for some r ∈ Z and γ = lch i ( P ( ww λ )) .Since b r + γ − a r ≥ n − l + 1 , we have a r ≤ b r + γ − n + l − b r + γ − l + l − ≤ b r + γ − . Let k be the largest index with a k < b r + γ . Therefore a r + n − l + 2 ≤ a k +1 < . . . < a r − l ≤ a r + n − . Claim: b k + γ ≥ a r + n − . If b k + γ ≤ a r + n − , then a r + n − l + 1 ≤ b r + γ < . . . < b k + γ ≤ a r + n − . So we get l different integers b r + γ , . . . , b k + γ , a k +1 , . . . , a r − l in a size l − interval [ a r + n − l + 1 , a r + n − , which is a contradiction, hence the claim is proved.Now we have a new matching a x < b x + γ − for all x ∈ Z , contrary to the definition of localcharge and property (3). (cid:3) Proposition 3.4.
The map F − → { X ∈ T ( λ ) | m | charge( X ) } via w P ( ww λ ) is a bijection.Proof. For any w ∈ F − , Lemma 2.5 tells P ( ww λ ) i = n w ( i ) , w ( i + m ) , . . . , w ( i + m ( l − o for i ∈ [ m ] . By the definition of Diff i and n P i =1 w ( i ) = n ( n +1)2 we know m divides Diff ( w ) + 2 Diff ( w ) + . . . + ( m −
1) Diff m − ( w ) . Therefore m | charge( P ( ww λ )) and the map is well-defined.We construct its inverse as follows. Pick a tabloid P ∈ T ( λ ) with m | charge( P ) , and let e = m − X i =1 lch i ( P ) − charge( P ) m . Define w to be w ( i + jm ) = P i,j +1 − e + P i − α =1 lch α ( P ) for i ∈ [ m ] , j ∈ [0 , l − . It can be checked explicitly that w ∈ F − and the procedure givesthe inverse of the map in the statement. (cid:3) Theorem 3.5. F − · w λ = LKC w λ , or equivalently, F = w λ · RKC w λ . Proof.
From Proposition 3.4, both sets F − · w λ and LKC w λ have the same cardinality so wewill only show LKC w λ ⊂ F − · w λ . Since w λ ∈ F − · w λ ∩ LKC w λ , it suffices to show that forany w ∈ F − ∩ LKC w λ · w λ , and s k ww λ ∼ LKC ww λ , there is s k w ∈ F − . Suppose k lies in row i of P ( ww λ ) and k + 1 lies in row i ′ .Case 1: i = i ′ ± . Then monotone conditions (1) (2) (in Definition 3.2) of F − are bepreserved when multiplying s k on the left. Then P ( ww λ ) and P ( s k ww λ ) can be computed irectly from Lemma 2.5 and we have P ( s k ww λ ) = s k P ( ww λ ) . If k = 0 , both Diff i and lch i will not change so (3) is satisfied as well. In case k = 0 , l − X j =0 (cid:24) s k w ( i + jm ) n (cid:25) = l − X j =0 (cid:24) w ( i + jm ) n (cid:25) + 1 , l − X j =0 (cid:24) s k w ( i ′ + jm ) n (cid:25) = l − X j =0 (cid:24) w ( i ′ + jm ) n (cid:25) − . Also we have lch i − ( P ( s k ww λ )) = lch i − ( P ( ww λ )) + 1 , lch i ( P ( s k ww λ )) = lch i ( P ( ww λ )) − , lch i ′ − ( P ( s k ww λ )) = lch i ′ − ( P ( ww λ )) − , lch i ′ ( P ( s k ww λ )) = lch i ′ ( P ( ww λ )) + 1 . Hence
Diff j ( s k w ) = lch j ( P ( s k ww λ )) .Case 2: i = i ′ ± . Let w ( i + am ) ≡ k (mod n ) and w ( i ′ + bm ) ≡ k + 1 (mod n ) .Case 2.1: a = b. Clearly s k w still satisfies monotone conditions (1) (2) and P ( s k ww λ ) = s k P ( ww λ ) . When k = 0 , Diff and lch do not change so (3) is satisfied as well. So we onlyneed to consider k = 0 . If in addition i ′ = i + 1 , lch i − ( P ( s k ww λ )) = lch i − ( P ( ww λ )) + 1 , lch i ( P ( s k ww λ )) = lch i ( P ( ww λ )) − , lch i +1 ( P ( s k ww λ )) = lch i +1 ( P ( ww λ )) + 1 , so (3) is again satisfied. i ′ = i − is similar.Case 2.2: a = b . s k w satisfies (1) trivially.Case 2.2.1: i ′ = i + 1 . By Lemma 3.3, < w ( i + 1 + am ) − w ( i + am ) ≤ n − l , so we have w ( i + 1 + am ) − w ( i + am ) = 1 , and ww λ (( a + 1) m − i ) − ww λ (( a + 1) m + 1 − i ) = 1 . Thenby definition of Knuth move, s k ww λ LKC ww λ , so this case cannot happen.Case 2.2.2: i = i ′ + 1 . Again by Lemma 3.3, there is < w ( i + 1 + am ) − w ( i + am ) ≤ n − l ,hence we know l = 1 , a = 0 and w ( i + 1) − w ( i ) = n − . Suppose ww λ ( n − i ) = k + αn forsome α ∈ Z , then ww λ ( n − i + 1) = k + 1 + ( α − n , and ww λ (2 n − i + 1) = k + 1 + αn .Because ww λ (1) > ww λ (2) > . . . > ww λ ( n ) , and (( ww λ ) − ( k − αn ) , k − αn ) or (( ww λ ) − ( k + 2 + αn ) , k + 2 + αn ) is ( n, n ) -translate of one of { ( i, ww λ ( i )) } ni =1 , hence cannotlie between row n − i and n − i + 1 , which contradicts s k ww λ ∼ LKC ww λ . (cid:3) Corollary 3.6.
Let w ∈ F − . Then P ( ww λ ) = w ( T λ ) , Q ( ww λ ) = T λ and ˜ ρ ( ww λ ) is aconstant vector with value − m − X i =1 lch i ( P ( ww λ )) + charge( P ( ww λ )) m = l − X j =0 (cid:24) w (1 + jm ) n (cid:25) − l. Proof.
From Theorem 3.5 we know ww λ ∈ F − · w λ = LKC w λ . And by Lemma 2.5, we have ˜ ρ i ( ww λ ) = l − X j =0 (cid:24) w ( i + jm ) n (cid:25) − l − i − X j =1 lch j ( P ( ww λ )) . enote P = P ( ww λ ) for simplicity. From the construction in Proposition 3.4, we have ˜ ρ i ( ww λ ) = l − X j =0 & P i,j +1 − e + P i − j =1 lch j ( P ) n ' − l − i − X j =1 lch j ( P ) = − e = − m − X i =1 lch i ( P ) + charge( P ) m . (cid:3) Corollary 3.7. φ m (LKC w λ ) = LKC w λ . Proof.
By definition of φ , φ m ( w ) = [ w (1 + ( l − m ) + m − n, w (2 + ( l − m ) + m − n, . . . , w ( n ) + m − n,w (1) + m, w (2) + m, . . . , w ( m ) + m, . . . , w (( l − m ) + m ] . And it can be checked easily that φ m ( w ) ∈ LKC w λ if w ∈ LKC w λ using the three conditionsof F and Theorem 3.5. (cid:3) We now study the explicit form of the right (resp. left) cells in C λ and relate them to theKnuth classes. In particular, we show that w λ · R T λ equals the set of all affine permutationssatisfying only the first two conditions of the fundamental box F (in Definition 3.2) in thefollowing proposition. Proposition 3.8.
The left cell L T λ containing w λ is the collection of all affine permutationssatisfying: (1) w ( i ) < w ( i + m ) < . . . < w ( i + m ( l − < w ( i ) + n for i ∈ [ m ] ; (2) w (1 + ( j − m ) > w (2 + ( j − m ) > . . . > w ( jm ) for j ∈ [ l ] .Proof. For simplicity, we denote the collection satisfying the two conditions above as C .By affine matrix ball construction, we know the affine permutations in C has recordingtabloid T λ , hence lying in L T λ .And by [19], any left cell is left-connected, hence it suffices to show that if w ∈ C and s k w ∈ L T λ , then s k w ∈ C . Suppose on the contrary that s k w / ∈ C , and suppose w ( i + jm ) ≡ k (mod n ) , w ( i ′ + j ′ m ) ≡ k + 1 (mod n ) for some i, i ′ ∈ [ m ] and j, j ′ ∈ [0 , l − .Case 1: j = j ′ . Then if in addition i < i ′ , or i > i ′ , w ( i ′ + j ′ m ) − w ( i + jm ) ≥ n + 1 , we have s k w ∈ C . It remains to consider the case j = j ′ , i = i ′ + 1 , and w ( i ′ + j ′ m ) − w ( i + jm ) = 1 . But in this situation the matrix balls ( i − jm, w ( i + jm )) , ( i + jm, w ( i + jm ) + 1) can becontained in a stream with density l + 1 , so λ ( s k w ) ≥ l + 1 , which contradicts s k w ∈ L T λ . Case 2: j = j ′ and i = i ′ , then clearly s k w ∈ C . Case 3: j < j ′ and i = i ′ . Since w ( i ) < w ( i + m ) < . . . < w ( i + ( l − m ) < w ( i ) + n, we know j ′ = j + 1 and w ( i + ( j + 1) m ) − w ( i + jm ) = 1 . In this situation, we have ananti-stream of m + 1 matrix balls in s k w : (1 + jm, w (1 + jm )) ≥ SW . . . ≥ SW ( i + jm, w ( i + jm ) + 1) ≥ SW ( i + ( j + 1) m, w ( i + jm )) ≥ SW ( i + 1 + ( j + 1) m, w ( i + 1 + ( j + 1) m )) ≥ SW . . . ≥ SW (( j + 2) m, w (( j + 2) m ) . Therefore λ T1 ( s k w ) ≥ m + 1 , which is also a contradiction.Case 4: j > j ′ and i = i ′ . This is the same as Case 3 after applying the rotation map φ . (cid:3) roposition 3.9. (cid:8) φ k ( w · R T λ ) = R φ k ( w ( T λ )) (cid:12)(cid:12) k ∈ [0 , m − , w ∈ F − (cid:9) is the collection of all right cells in C λ . Moreover, ℓ ( ww λ w ′ ) = ℓ ( w ) + ℓ ( w λ ) + ℓ ( w ′ ) for w ∈ F − and w λ w ′ ∈ R T λ . Proof.
For any w ∈ F − = LKC w λ · w λ , w λ w ′ ∈ R T λ , we claim:(1) ww λ and ww λ w ′ are in the same right cell;(2) w λ w ′ ∼ LKC ww λ w ′ .Since any right cell is right-connected [19], we could find a path (right multiplication by sim-ple reflections) in R T λ connecting w λ and w λ w ′ . Similarly, we have a path (left Knuth moves)in LKC w λ connecting w λ and ww λ . Hence by induction, it suffices to assume ww λ w ′ , ww λ w ′ s i being in the same right cell and ww λ w ′ ∼ LKC s j ww λ w ′ , and prove s j ww λ w ′ , s j ww λ w ′ s i beingin the same right cell and ww λ w ′ s i ∼ LKC s j ww λ w ′ s i . ww λ w ′ ww λ w ′ s i s j ww λ w ′ s j ww λ w ′ s i · s i s j · s j ·· s i Denote P = P ( ww λ w ′ ) and Q = Q ( ww λ w ′ ) . Since ww λ w ′ ∼ LKC s j ww λ w ′ , we know P ( s j ww λ w ′ ) = P ′ , and Q ( s j ww λ w ′ ) = Q where P and P ′ differs by a Knuth move.Since ww λ w ′ , ww λ w ′ s i are in the same right cell, we have P ( ww λ w ′ s i ) = P , and denote Q ( ww λ w ′ s i ) = Q ′ . By [4, Proposition 3.23], we know there is a unique affine permutationwith insertion tabloid P ′ and recording tabloid Q ′ that is related to ww λ w ′ s i by a left Knuthmove. Suppose this affine permutation is s j ′ ww λ w ′ s i . And we claim j = j ′ .Without loss of generality, suppose for some α ∈ [ n ] , α ∈ τ ( P ) , α + 1 / ∈ τ ( P ) , α + 1 ∈ τ ( P ′ ) , α / ∈ τ ( P ′ ) . Hence by Proposition 2.13, ( ww λ w ′ ) − ( α + 1) is smallest one among ( ww λ w ′ ) − ( α ) , ( ww λ w ′ ) − ( α + 1) , and ( ww λ w ′ ) − ( α + 2) . Similarly, ( ww λ w ′ ) − ( α + 1) issmallest one among ( ww λ w ′ s i ) − ( α ) , ( ww λ w ′ s i ) − ( α + 1) , and ( ww λ w ′ s i ) − ( α + 2) . The onlypossibilities that j = j ′ are the following two cases:Case 1: ( ww λ w ′ ) − ( α +1) < ( ww λ w ′ ) − ( α ) < ( ww λ w ′ ) − ( α +2) , but ( ww λ w ′ s i ) − ( α +1) < ( ww λ w ′ s i ) − ( α + 2) < ( ww λ w ′ s i ) − ( α ) , so j = α + 1 and j ′ = α . This happens iff i = ( ww λ w ′ ) − ( α ) = ( ww λ w ′ ) − ( α + 2) − . Since ww λ w ′ ∈ C l m , we know each matrix ball in B ww λ w ′ , and in particular (( ww λ w ′ ) − ( α +2) , α + 2) = ( i + 1 , α + 2) , is contained in an anti-stream of density m . But if we replace ( i + 1 , α + 2) with ( i + 1 , α ) and ( i, α + 2) , we obtain an anti-stream of density m + 1 in B ww λ w ′ s i , which contradicts ww λ w ′ s i and ww λ w ′ being in the same right cell.Case 2: ( ww λ w ′ ) − ( α +1) < ( ww λ w ′ ) − ( α +2) < ( ww λ w ′ ) − ( α ) , but ( ww λ w ′ s i ) − ( α +1) < ( ww λ w ′ s i ) − ( α ) < ( ww λ w ′ s i ) − ( α + 2) , so j = α and j ′ = α + 1 . This happens iff i = ( ww λ w ′ ) − ( α + 2) = ( ww λ w ′ ) − ( α ) − . Similar to the previous case, since ww λ w ′ ∈ C l m , we know each matrix ball in B ww λ w ′ , andin particular (( ww λ w ′ ) − ( α ) , α ) = ( i + 1 , α ) , is contained in a stream of density l . But if we eplace ( i + 1 , α ) with ( i, α ) and ( i + 1 , α + 2) , we obtain a stream of density l + 1 in B ww λ w ′ s i ,which contradicts ww λ w ′ s i and ww λ w ′ being in the same right cell.So both cases cannot happen and we arrived at the claim j = j ′ . By Proposition 2.13 and the second length formula in Lemma 2.1, we know that ℓ ( s j ww λ w ′ ) − ℓ ( ww λ w ′ ) = ℓ ( s j ww λ w ′ s i ) − ℓ ( ww λ w ′ s i ) . Therefore by induction there is: ℓ ( ww λ w ′ ) = ℓ ( ww λ ) + ℓ ( w λ w ′ ) − ℓ ( w λ ) = ℓ ( w ) + ℓ ( w λ ) + ℓ ( w ′ ) and the last equality is due to the monotone condition in Proposition 3.8 and the first lengthformula in Lemma 2.1.Now from the claim, we know w · R T λ is contained in the right cell R w ( T λ ) . Moreover, wehave a map R T λ → R w ( T λ ) by x wx , and in fact left multiplication by w − gives an inverseof this map, hence w · R T λ = R w ( T λ ) . Applying rotations we get φ k ( w · R T λ ) = R φ k ( w ( T λ )) .Lemma 3.1 indicates these are all the right cells in C λ . (cid:3) For k ∈ [ m − , let w ( k ) be the following affine permutation: w ( k ) = φ k ([ m − k + 1 , m − k + 2 , . . . , m, , , . . . , m − k, m − k + 1 , m − k + 2 , . . . , m, m, m, . . . , m − k,. . .n − k + 1 , n − k + 2 , . . . , n, l − m, l − m, . . . , n − k ]) . Lemma 3.10. ( w ( k ) ) − ∈ L T λ · w λ , but ( w ( k ) ) − / ∈ F − . And there exists some w ∈ F − andsome i ∈ [0 , n − , such that ( w ( k ) ) − = s i w .Proof. By definition, ( w ( k ) ) − = [ 1 − m + k, − m + k, . . . , k − m, k + 1 , k + 2 , . . . , m + k, k, k, . . . , k, k + 1 + m, k + 2 + m, m + k,. . . k + ( l − m, k + ( l − m, . . . , k + ( l − m, k + 1 + ( l − m, k + 2 + ( l − m, . . . , ml + k ] It can be checked directly that ( w ( k ) ) − ∈ L T λ · w λ , Diff i (( w ( k ) ) − ) = lch i ( P (( w ( k ) ) − w λ )) when i = k , but Diff k (( w ( k ) ) − ) = lch k ( P (( w ( k ) ) − w λ )) + 1 . So ( w ( k ) ) − / ∈ F − . There aredifferent ways to find a pair of required w and i . One way is to take i = k, m > km, m = 2 k k − m, m < k . And one can check directly that s i ( w ( k ) ) − ∈ F − . (cid:3) Proposition 3.11.
For any w ∈ L T λ · w λ , either w ∈ F − , or there exists some k ∈ [ m − and w ′ ∈ L T λ · w λ , such that w = φ k ( w ′ )( w ( k ) ) − and Diff( w ′ ) − lch( P ( w ′ w λ )) < Diff( w ′ ) − lch( P ( w ′ w λ )) . Moreover, ℓ ( w ) = ℓ ( w ′ ) + ℓ ( w ( k ) ) . roof. For any given w ∈ L T λ · w λ \ F − , there exists some k ∈ [ m − such that Diff k ( w ) > lch k ( P ( ww λ )) . Let w ′ = φ − k ( ww ( k ) ) . Direct computation gives that w ′ equals [ w ( m +1) − k, w ( m +2) − k, . . . , w ( m + k ) − k, w (1+ k ) − k, w (2+ k ) − k, . . . , w ( m ) − k,w (2 m +1) − k, w (2 m +2) − k, . . . , w (2 m + k ) − k, w (1+ k + m ) − k, w (2+ k + m ) − k, . . . , w (2 m ) − k,. . .w (1)+ n − k, w (2)+ n − k, . . . , w ( k )+ n − k,w (1+ k +( l − m ) − k, w (2+ k +( l − m ) − k, . . . , w ( n ) − k ] . Since
Diff k ( w ) > lch k ( P ( ww λ )) , we have w (( j + 1) m + k ) − k > w (1 + k + jm ) − k for all j ∈ [0 , l − , and clearly w ′ preserves monotone conditions from w . Moreover, Diff i ( w ′ ) − lch i ( P ( w ′ w λ )) = (cid:26) Diff i ( w ) − lch i ( P ( ww λ )) , i = k ;Diff k ( w ) − lch k ( P ( ww λ )) − , i = k. By Shi’s length formula in Lemma 2.1, ℓ ( w ) = ℓ ( φ − k ( w )) = ℓ ( w ′ ) + l ( m − k ) k = ℓ ( w ′ ) + ℓ ( w ( k ) ) . (cid:3) An immediate corollary is the following:
Corollary 3.12.
For any w ∈ w λ · R T λ , we have the following expression: w = w ( k ) φ k ( w ( k ) φ k ( · · · w ( k ε ) φ k ε ( w ′ ) · · · )) where w ′ ∈ F and { k , . . . , k ε } = { d , d , . . . , ( m − d m − } as a multi-set (the order of k i ’sdoes not matter) and d j = Diff j ( w − ) − lch j ( P ( w − w λ )) for j ∈ [ m − . Proposition 3.13.
Let w ∈ w λ · R T λ and { j < j < . . . < j s } = (cid:8) j ∈ [ m − | Diff j ( w − ) > lch j ( w − w λ ) (cid:9) . Then the number of elements in Ψ − ( w λ w ) is − ( w λ w ) = − ( w − w λ ) = (cid:18) mj , j − j , . . . , j s − j s − , m − j s (cid:19) . Proof.
Since ρ i ( w − w λ ) = l − X j =0 (cid:24) w − ( i + jm ) n (cid:25) − l, we have ˜ ρ i +1 ( w − w λ ) − ˜ ρ i ( w − w λ ) = l − X j =0 (cid:24) w − ( i + 1 + jm ) n (cid:25) − l − X j =0 (cid:24) w − ( i + jm ) n (cid:25) − lch i ( P ( w − w λ ))= Diff i ( w − w λ ) − lch i ( P ( w − w λ )) . Therefore, ˜ ρ ( w − w λ ) = . . . = ˜ ρ j ( w − w λ ) < ˜ ρ j +1 ( w − w λ ) = . . . = ˜ ρ j ( w − w λ ) < ˜ ρ j +1 ( w − w λ ) = . . .. . . = ˜ ρ j s ( w − w λ ) < ˜ ρ j s +1 ( w − w λ ) = . . . = ˜ ρ m ( w − w λ ) . ence by Theorem 2.9, − ( w λ w ) = − ( w − w λ ) = (cid:18) mj , j − j , . . . , j s − j s − , m − j s (cid:19) . (cid:3) Affine Springer fibers and their geometry
In this section we will introduce the geometric spaces that will appear in this paper, andexplain some of their basic properties.Denote by K := C (( t )) and O := C [[ t ]] the Laurent power series and power series algebraswith complex coefficients respectively.Let G be a reductive group, B ⊂ G a Borel subgroup and T ⊂ B a maximal torus . Denoteby g the Lie algebra of G . The root system of G of roots, weights, coroots and coweightsis given by ( R, X , R ∨ , X ∨ ) . Further the choice of B , gives a choice of positive roots R + . Wealso have the set of affine roots , given by R aff := R × Z δ ∪ { } × Z δ , where δ is the generatorof the affine direction. Associated to this, we have the Weyl group W , the affine Weyl group f W := W ⋉ Z R ∨ and the extended affine Weyl group f W ext := W ⋉ X ∨ . In the case of a simplyconnected group, we have Z R ∨ = X ∨ and thus the affine Weyl group and the extended affineWeyl group agree.We can now construct an Iwahori subgroup I of G ( O ) ⊂ G ( K ) via the pullback diagram I G ( O ) B G t =0 With this, we are ready to define the affine flag variety F l as the ind-scheme whose closedpoints are given by the quotient G ( K ) /I . We will only consider the properties of the reducedstructure of this space so we omit the details of its scheme structure.The affine flag variety has a Schubert decomposition into locally closed subsets given by I -orbits labeled by f W ext . This is given by a natural inclusion f W ext = N G ( K ) ( T ( K )) /T ( O ) ֒ → F l .Considering the I -orbits we get G w ∈ f W ext IwI/I.
This decomposition can also be understood as a decomposition of F l × F l into G ( K ) -orbits. These again are labeled by f W ext , by considering the G ( K ) -orbit of (1 , w ) . For twopoints x, y ∈ F l , we say they are in relative position r ( x, y ) = w , if ( x, y ) ∈ F l × F l is inthe G ( K ) -orbit labeled by w . Note that we have r ( x, y ) = r ( y, x ) − and r ( gx, gy ) = r ( x, y ) for any g ∈ G ( K ) .This decomposition also induces a partial order into f W ext , known as the Bruhat order ,given by w ≤ w ′ if IwI/I ⊂ Iw ′ I/I or equivalently G ( K )(1 , w ) ⊂ G ( K )(1 , w ′ ) .We can now extend the notion of relative position to pairs of irreducible subvarieties of F l . Namely let X, Y ⊂ F l be two irreducible subvarieties. Then X × Y ⊂ F l × F l has astratification into locally closed subsets given by intersections with G ( K ) -orbits. As X × Y is irreducible there is a unique G ( K ) -orbit, say labeled by w , that intersects X × Y in anopen subset. We then denote r ( X, Y ) = w and say X and Y are in relative position w . Notethat as a consequence X × Y ⊂ G ( K )(1 , w ) . Thus for any pair of points x ∈ X , y ∈ Y , we ave r ( x, y ) ≤ r ( X, Y ) and generically this is an equality. The same properties as above arethus easy to see, i.e. r ( X, Y ) = r ( Y, X ) − and r ( gX, gY ) = r ( X, Y ) for any g ∈ G ( K ) .We also have a similar definition of relative position for the finite flag variety G/B interms of B -orbits of G/B and G -orbits of G/B × G/B given by elements of the Weyl group W , but we omit the details as it is essentially the same construction as above.We can now introduce the affine Springer fiber associated to an element γ ∈ g ( K ) followingthe work of [14]. This is a subscheme F l γ ⊂ F l with closed points given by F l γ := { gI ∈ F l | γ ∈ g Lie ( I ) } . The space F l has automorphisms given by left multiplication with elements of G ( K ) .These automorphisms preserve F l γ if they centralize γ .We will now focus on the G = SL n . Recall that for SL n , X ∨ := { µ ∈ Z n | P i µ i = 0 } .The roots are given by R = { e i − e j | i, j ∈ [ n ] } and R + = { e i − e j | i ≤ j } . The Weyl groupfor SL n is W = S n and the affine Weyl group is f W = f S n . Since SL n is simply connected,the extended affine Weyl group is the same as the affine Weyl group as stated above.We consider the affine Springer fiber for SL n and γ = N ∈ sl n ( O ) a generic lift of anilpotent of sl n corresponding to the partition ( l m ) . This is the nil-element considered inLusztig’s conjecture [18] in the case of the nilpotent corresponding to the partition ( l m ) . Wecan consider the explicit choice given by N = I . . . ... . . .
00 0 . . . I th . . . . Here the blocks are m × m matrices, I is the identity matrix and h is a regular semisimpleelement, which without loss of generality can be taken to be a diagonal matrix with distinctnon-zero eigenvalues.Note that N is conjugate to the following block diagonal matrix J . . . J . . . ... . . . ... . . . J m where the l × l diagonal block matrices are J i := . . .
00 0 1 . . . ... . . . ... . . . th i . . . . This matrix is centralized by the block diagonal matrices f i , where all the blocks arereplaced with the identity except J i . After conjugating this gives matrices f ′ i centralizing N .Note that these matrices are elements of GL n ( K ) , but not SL n ( K ) , in fact their determinanthas valuation . We now introduce a more explicit description of F l for SL n and then seehow these matrices give automorphisms of F l N . o do this we introduce the notion of a O -lattice V inside K n , as a O -submodule, suchthat V ∼ = O n as an O -module. Note that V n K n = K , and thus V n V ⊂ K is a rank free O -module and thus we must have V n V = t k O ⊂ K for some k .With this description we have that for SL n : F l = { V ⊂ V ⊂ . . . V n − ⊂ t − V | V i is a lattice in K n and n ^ V i = t − i O} . We can now see that V n gV = det( g ) V n V and thus we have the automorphism of F l givenby ( F i ( V j )) k = ( f ′ i ) − V k − . This has the correct determinant and further as f ′ i centralize N this induces automorphisms F i on F l N .We will consider automorphisms F c := F c . . . F c m m , for c an m -tuple. This is well-defined as the F i clearly commute with each other. Note that if P c i = 0 we have ( f ′ ) c =( f ′ ) − c . . . ( f ′ m ) − c m , which is indeed an element of SL n ( K ) and so F c is just multiplication byan element in SL n ( K ) . We refer to these transformations as the translations and the groupof translations is denoted by Λ .We introduce the parahoric e P generated by the Iwahori I and the simple reflections s i such that i mod m ) . We will also need the parabolic subgroup W P ⊂ f W correspondingto this parahoric, i.e. the subgroup generated by the reflections s i such that i mod m ) as above. For this parabolic subgroup W P , the maximal element is w P = w λ as defined afterTheorem 2.4.We can consider the family of spaces F l N over the space S rs of regular semisimple diagonal m × m -matrices h with non-zero eigenvalues, given by varying N in the obvious way. This isa fiber bundle where each fiber is homeomorphic and thus we can consider the monodromyaction on components. Note that the monodromy acts on the points of f W through S m ֒ → ( S m ) l ∼ = W P ⊂ f W . We will use this to get an action of S m on the set of components.5. Components of affine Springer fiber of rectangular type
In this section we study the points of f W appearing in F l N as well as the intersectionwith the orbits of the parahoric e P introduced in the previous section. To study this werequire certain equations that recur throughout the paper. To introduce them, recall thecoweight lattice X ∨ with the action of f W . Further consider the set of connected componentsof X ∨ ⊗ R r ∪ α {h α, µ i ∈ Z } . The closures of these are known as the set of alcoves A and wehave that f W acts on A . Consider the fundamental alcove given by A := { ≤ h α, µ i ≤ , ∀ α ∈ R + } . Then we have a bijection f W ∼ = A given by w A w := w ( A ) , i.e. by acting on thefundamental alcove A .The following three sets of equations on alcoves are used throughout this section:(1) ≤ h e i + am − e i + bm , A w i ≤ for i ∈ [ m ] and ≤ a < b ≤ l − , (2) h e i + jm − e i +1+ jm , A w i ≥ for i ∈ [ m − and j ∈ [0 , l − , i ( i ∈ [ m − ) At least one of the following is satisfied: ≤ h e i +1 − e i + m , A w i ≤ , ≤ h e i +1+ m − e i +2 m , A w i ≤ ,. . . ≤ h e i +1+( l − m − e i +( l − m , A w i ≤ , ≤ h e i − e i +1+( l − m , A w i ≤ . We say A w satisfies equation (3) if it satisfies (3) i for all i ∈ [ m − . In fact these three sets of equations give an equivalent definition of the fundamental boxby the following proposition.
Proposition 5.1. w ∈ F if and only if the corresponding alcove A w satisfies equations (1),(2) and (3).Proof. Since A w = w ( A ) = (cid:8) µ | µ w ( n ) < µ w ( n − < . . . < µ w (1) < µ w ( n ) + 1 (cid:9) , equations (1) and (2) are equivalent to first two conditions in Definition 3.2, and we showthe third are the same as well.Let w ∈ F , and suppose on the contrary that w does not satisfy equation (3). Then fromthe monotone conditions of w − , there exists some i ∈ [ m − : w − ( i + 1) > w − ( i + m ) ,w − ( i + 1 + m ) > w − ( i + 2 m ) ,. . .w − ( i + 1 + ( l − m ) > w − ( i + ( l − m ) ,w − ( i + 1 + ( l − m ) > w − ( i ) + n. But these inequalities imply lch i ( P ( w − w λ )) < Diff i ( w − ) , which contradicts w ∈ F .Now for any w satisfying all three sets of equations, we know Diff i ( w − ) ≥ lch i ( P ( w − w λ )) for all i ∈ [ m − by (1) and (2). If for some i , Diff i ( w − ) ≥ lch i ( P ( w − w λ )) + 1 , then the l inequalities above holds, which is contradictory to equation (3) i . (cid:3) We now begin by understanding the points of f W that are contained in F l N . Lemma 5.2. w ∈ f W ∩ F l N ⇔ the alcove A w satisfies equation (1) .Proof. w ∈ f W ∩ F l N ⇔ w N ∈ Lie ( I ) . Note that N has a non-zero entry in the weight spaces e i − e i + m for i + m ≤ ml and e i +( l − m − e i + δ for i ≤ m .Thus from the above we have w ∈ F l N ⇔ w − ( α ) is a positive root α ∈ R aff for the rootspaces with non-zero entries in N . These conditions translate to the conditions ≤ h e i − e j , A w i ≤ for i ≡ j ( mod m ) and we thus get the elements in f W ∩F l N are exactly described by equation (1) . (cid:3) o understand the components better, we consider e P -orbits. These turn out to be adisjoint union of smooth open subsets of a number of components. To prove this statementwe follow ideas of [9].Before we start we introduce some torus actions on F l N . First consider T ⊂ SL n thediagonal torus. We can construct a subtorus S = { diag ( µ i ) | µ i = µ j if i ≡ j ( mod m ) } . Since S commutes with N , S acts on F l N .We also have an action of G m on F l by loop rotations, i.e. by scaling the uniformizer t ofthe algebra O of power series.We can then consider the following cocharacter µ : G m → T × G m described by µ ( x ) =( diag ( µ i ( x )) , δ ( x )) , such that µ i = x − ⌊ im ⌋ and δ ( h ) = x − l . We can check that this acts byscaling N and thus acts on F l N .Consider the Lie algebra e p of e P . Then the action through µ gives a grading on e p , withnon-negative weights. Denote the graded pieces by e p k and the filtered pieces e p >k = ⊕ e p k .The graded part is the Levi of e p , which we denote by e l := e p and by e L the correspondingLevi subgroup. Further note that the S acts on e l and there is a cocharacter of S such thatLie ( I ) ∩ e l is exactly the non-negative weight spaces.Before stating the following Lemma for w satisfying equation (2) , we introduce the notation Y ◦ w = e P wI/I ∩ F l N and Y w = e P wI/I ∩ F l N . Note that if we consider w satisfying equation (2) , we do indeed get all the e P -orbits. Lemma 5.3. If Y w is non-empty, then W P w ∩ F l N is non-empty.Proof. Consider the cocharacter µ : G m → T × G m defined above.Note that with respect to this cocharacter e P has all non-negative root spaces. Thus weget e P -orbits are contracted by the action through λ to the orbit of e L on the points of f W .We thus must have if e P -orbit intersects F l N then the e L -orbit on the points of f W alsointersects F l N . Note that S acts on e L and so acts on the e L -orbits. Then we can consider acocharacter of S such that the non-negative weights on e l are exactly given by the intersectionof e l with Lie ( I ) . Under this action the e L -orbit on the points of f W contracts to one of thosepoints.We thus have that if e P -orbit intersects F l N , then it intersects it at a point of f W of thatorbit. Theses points are exactly as described in the statement of the lemma and thus theresult follows. (cid:3) Corollary 5.4.
All the non-empty Y w are given by w satisfying equations (1) and (2) (i.e. w ∈ w λ · R T λ ).Proof. Note that by the previous lemma Y w is non-empty if and only if some element in W P w satisfies equation (1) .It follows from Lemma 2.19 that if w ′ satisfies equation (1) then there is an element in W P w ′ satisfying equations (1) and (2) . (cid:3) It follows from this Corollary and Proposition 3.8 that F l N = [ w ∈ w λ · R Tλ Y w . emma 5.5. Y ◦ w is always smooth and equidimensional of dimension dim( e P /I ) , when it isnon-empty.Proof. The following proof follows [9].We want to understand the tangent spaces to Y ◦ w for w satisfying equation (1) .Note that the e P -orbit at w is isomorphic to e P / e P ∩ w − I. We have the tangent bundle on this space given by e p / e p ∩ w − g − Lie ( I ) at the point gwI/I .Over the intersection with F l N , we have the mapad ( N ) : e p / e p ∩ w − g − Lie ( I ) → e p / e p ∩ w − g − Lie ( I ) given by the adjoint action. Note that the image is in the nilpotent radical u e p of e p . In factthe map ad ( N ) : e p / e p ∩ w − g − Lie ( I ) → u e p /u e p ∩ w − g − Lie ( I ) is surjective.The tangent space at a point in Y ◦ w is given by the kernel of the above map. It follows,as this map is surjective, that the dimension of all tangent spaces is the same and thus theintersection is smooth.To compute the dimension of each component, we just need to compute the tangent spaceat any point, which is given by the kernel of the above map of vector bundles. But note thatthe dimension of this is just the codimension of the above vector spaces, thus the dimensionof the tangent space is the codimension of e p / e p ∩ w − Lie ( I ) and u e p /u e p ∩ w − Lie ( I ) . This isexactly given by the dimension of e l / e l ∩ w − Lie ( I ) , where e l is the Lie algebra of the Levisubgroup e L .The intersection e l ∩ w − Lie ( I ) always gives a Borel subalgebra of e l and hence the abovespace is always of dimension dim( e P /I ) as required. (cid:3) It follows from Lemma 5.5 that the intersection with e P -orbits give disjoint smooth opensets of components. To understand the components we want to identify precisely when theintersection is irreducible. We state the exact conditions in the following lemma, but firstwe introduce some geometric spaces known as Hessenberg varieties .Consider G a reductive group and B a Borel subgroup. Further consider a G -representation V and a B stable subrepresentation W ⊂ V (this notation should not cause any confusionwith the notation for the Weyl group). Further consider a vector v ∈ V . Then we canconstruct the Hessenberg variety H Wv := { gB ∈ G/B | v ∈ gW ⊂ V } . With this we can state the lemma.
Lemma 5.6.
For A w satisfying equations (1) and (2) , Y w is irreducible if and only if w ∈ F ,i.e. also satisfies equations (3) .Proof. Consider the filtration on e P induced by the filtration e p >k , which we denote by e P >k .We can use this filtration to construct the following quotients e P >k \ e P / e P ∩ w − I factoring the map e P / e P ∩ w − I → e P /I . his induces a similar sequence of maps on the intersection e P wI/I ∩ F l N . Each of thesequence of maps induced on this space is an affine space bundle over its image. This followsfrom the results from [9].Further from [9] we also get explicitly the image on e P /I . This image is given by aHessenberg variety. Namely consider the representation V = e p , which is a e l and also a e L -representation. Note that N ∈ e p and consider W w := w − Lie ( I ) ∩ e p for w satisfyingequations (1) and (2) . This is stable under the action of the Borel B e L ⊂ e L given by theimage of I .As proven in [9], the image is given by the Hessenberg variety H W w N . Note further it followsfrom [9] or from the above descriptions that H W w N is smooth. It thus follows that to proveirreducibility of the e P -orbit and hence of Y w it is equivalent to proving irreducibility of theHessenberg variety of H W w N and hence on the connectedness of this Hessenberg variety.Using the action by a cocharacter of S on this Hessenberg variety we get that B e L -orbits arethe attracting sets and that to prove connectedness we just need to prove that all the pointsin W P contained in this Hessenberg variety are indeed in the same connected component.We now give a more explicit description of the flag variety e P /I and of the above Hessenbergvariety. Note that the flag variety e P /I is just given by ( SL m /B ) l , so it is just given bycomplete flags of l distinct m -dimensional vector spaces. Let us denote them by V i for i = 1 , . . . , l .The representation V of e L can be described as the set ⊕ Hom( V i , V i +1 ) , where we interpret i +1 ( mod l ) . With this description and an appropriate choice of basis we can describe N ∈ V as the identity map between V i → V i +1 for i = 1 , . . . , l − and a diagonal map with distincteigenvalues for the map V l → V .With this construction it is clear that we can compose all the maps Hom( V i , V i +1 ) toget a map Hom( V k , V k ) . We thus get several maps from the Hessenberg variety H W w N tosome Hessenberg variety H Wh on SL m /B with the representation of End ( V k ) and the regularsemisimple endomorphism h , introduced above, as the vector v and some B stable subrepre-sentation W . These maps are given by projecting to the k -th factor of ( SL m /B ) l . It is easyto see that all the points of the Weyl group S m are in the image of this map.These Hessenberg varieties H h are studied in [2, 3]. It is proven there that for a B stablesubrepresentation Lie ( B ) ⊂ W ⊂ End ( V k ) the Hessenberg variety is connected if and onlyif all negative finite simple roots are contained in W . If H W w N is connected, then the imageHessenberg variety H Wh must be connected as well. The condition for W to satisfy that H Wh is connected, is equivalent to equation (3) for w .Thus w ∈ F is necessary and it remains to prove it is sufficient.We thus need to prove that the points in W P = ( S m ) l appearing in the Hessenberg varietyare all on the same component. To do this, we break it up in two steps. We prove in thefollowing Lemma 5.7 that the fibers over S m of the map to SL m /B above are connected.Note that the diagonal S m ֒ → ( S m ) l is always included in the Hessenberg variety andby Lemma 5.7 we have that every point of W P in our Hessenberg variety is in the sameconnected component as one of the diagonal ones. It remains to show that the diagonal onesare in the same connected component. To do this it is enough to proof that ( w ) and ( ws i ) fora simple reflection s i are in the same component. Without loss of generality we can assume w = id . onsider the SL α i ֒ → SL m the subgroup corresponding to the simple root α i . Let g ∈ SL α i . We can consider a subspace given by points ( x i ) ∈ ( SL m /B ) l where x i = gB or x i = hgB . Note that the flags ( G i ) corresponding to gB and hgB are given by G j = h e , . . . , e j i for j = i . Thus to check whether such a point lies in the space we only need to check theconditions for the i th vector space in each flag.Note that hG i ⊂ G i +1 and hG i − ⊂ G i . Thus the tuple ( x k ) satisfies the conditionsregardless of the choice gB or hgB for x k , unless one of the Hessenberg conditions is N G ki ⊂ G k +1 i . If we have this condition we must have x k = x k +1 in the case k = l and x l = gBx = hgB in the case k = l .Equation (3) i is equivalent to the condition N G ki ⊂ G k +1 i not being required for every k . Under those conditions it is easy to see that we can indeed choose ( x i ) satisfying allHessenberg conditions for any g ∈ SL α i . This gives a P inside the Hessenberg varietycontaining ( id ) and ( s i ) . It follows these two are in the same connected component and theresult follows. (cid:3) Lemma 5.7.
The fibers of the map of Hessenberg varieties H W w N → H h at S m are connected,when w satisfies equation (1) and (2) .Proof. Without loss of generality we assume we are projecting to the last copy of SL m /B in ( SL m /B ) l .We give certain conditions under which two elements of ( S m ) l in the fiber of the abovemap are in the same connected component of the Hessenberg variety H W w N . We will thenuse this construction to prove every point of S lm appearing in the Hessenberg variety in afixed fiber is in the same connected component. It follows from the discussion in the proofof Lemma 5.6 that this proves the fibers of the map are indeed connected.Consider the subgroup SL α ֒ → SL m corresponding to the positive root α .Consider two points ( w i ) , ( w ′ i ) ∈ ( S m ) l ∩ H W w N that satisfy w ′ i = ( s α w i , if k ≤ i ≤ k w i , otherwisefor some k < k . To see that these two points lie in the same connected component considerthe subspace of ( SL m /B ) l given by coordinates ( x i ) with x i = ( gw i B/B, if k ≤ i ≤ k w i B/B, otherwisefor g ∈ SL α .We check this is in the Hessenberg variety H W w N . The only conditions that need to bechecked are in the boundry cases k i . Note here that the condition at k only depends on theflags of w k and gw k +1 and the condition for N at i = l only depends on the relative position,i.e. it only depends on the Schubert cell in which the flag corresponding to w − k gw k +1 lies.Note that the action of the torus S makes the set of flags given by w − k gw k +1 an S -stable P with fixed points given by w − k w k +1 and w − k s α w k +1 . The Schubert cells are just attractingsets along some cocharacter of S . It follows that the relative position of w − k gw k +1 with theidentity is given by either w − k w k +1 or w − k s α w k +1 . By assumption both of these do satisfythe conditions of the Hessenberg variety and it thus follows that the above subspace givenby ( x i ) lie in the Hessenberg variety. This is a connected subvariety containing both ( w i ) and ( w ′ i ) , hence these two points are in the same connected component. ow we check that using the above condition, everything can be shown to be in the sameconnected component. To do this we will consider without loss of generality the fiber atthe identity of S m . Then we show every point is in the same connected component as thediagonal identity fixed point.To do this consider a point ( w i ) ∈ ( S m ) l ∩ H W w N . We prove that if ( w i ) = ( id ) there is apoint ( w ′ i ) ∈ ( S m ) l ∩ H W w N satisfying the conditions above for some positive root α and some k < k such that the length of ( w ′ i ) − w ′ i +1 is at most the length of w − i w i +1 and at leastone of them is strictly smaller, where here we consider i + 1 ( mod l ) . The result follows byinduction as w l = id by assumption.Note that after multiplying by s α we get ( w ′ i ) − w ′ i +1 = w − i w i +1 or ( w ′ i ) − w ′ i +1 = w − i s α w i +1 .We will want to check what the condition is for w − i s α w i +1 having smaller length than w − i w i +1 . This can be checked to happen exactly when one of w − i ( α ) and w − i +1 ( α ) is apositive root and the other is negative.Note that w − l ( α ) is positive, so writing the sign of w − i ( α ) in a string, we have a sequenceof + and − starting and ending with + . We can then choose a consecutive substring of all − ’s and multiply those elements by s α to get ( w ′ i ) . By the above description this reducesthe length of some products w − i w i +1 and leaves others unchanged.The only remaining thing to check is that ( w ′ i ) is indeed in the Hessenberg variety H W w N .The conditions of the Hessenberg variety are given by conditions on Hom( V k , V k +1 ) . Theseconditions only depend on w − i w i +1 . Further the conditions are given by some bounds inBruhat order of w − i w i +1 , but by construction ( w ′ i ) − w ′ i +1 is smaller in Bruhat order. Itfollows that this point also satisfies the conditions of the Hessenberg variety.To finish note that if there is no − ’s string in the sequence constructed above for anypositive root α , then we must have w i = id ∀ i and thus the the connectedness follows. (cid:3) We now are ready to give a formula for the relative position of Y w for w ∈ F to any Y w ′ .To introduce this we recall the notation w λ = w P ∈ W P for the longest element in the finiteparabolic group W P . Theorem 5.8. If w ∈ F and w ′ satisfies equations (1) and (2) (i.e. w ∈ w λ · R T λ ), then therelative position between Y w and any component of Y w ′ is given by w − w P w ′ .Proof. The relative position between the e P -orbits at w and w ′ is exactly given by r ( e P wI/I, e P w ′ I/I ) = w − w P w ′ , as the length of the product is the sum of the lengths. This follows from Proposition 3.9.Consider a component X of Y w ′ . It thus follows that we have a bound on relative positiongiven by r ( Y w , X ) ≤ r ( e P wI/I, e P w ′ I/I ) = w − w P w ′ .Now note that any component in Y w ′ contains a point of f W in the e P -orbit of w ′ , as wehave seen in the proof of Lemma 5.3. It thus follows for each component there exists an x ∈ W P such that the component contains xw ′ . Hence we have, that the relative positionbetween xw ′ and x is given by ( w ′ ) − .Now consider the relative position of x with Y w . For this it is enough to consider the e P -orbit at w . Further it is easy to see that the points in relative position w P w to x areprecisely the preimage under the map e P wI/I → e P /I of the points in relative positions w P to the point x . Thus to check that the relative position of x to Y w is w p w we just need tocheck that the relative position of x to the Hessenberg variety H W w N is w P . ecall that we have l projections π i : e P /I → SL m /B . Consider the preimage U i of thepoints in relative position w ∈ S m to π l ( x ) . The points in relative position w P to x is givenby the intersection of all U i .The Hessenberg variety for w ∈ F is irreducible by Lemma 5.6. It follows that if U i intersects non-trivially with H W w N for all i , then there is a point in H W w N in relative position w P to x , as non-empty open subsets of H W w N intersects in a non-empty subset. Recall thatwe contain the diagonal inclusion of S m → ( S m ) l in the Hessenberg variety H W w N . It is clearthat for each U i there exists one of these points contained in U i . It thus follows that therelative position of Y w to x is w − w P . Thus we get that the relative position of Y w and xw ′ is w − w P w ′ , thus the relative position of Y w and any components of Y w ′ is at least w − w P w ′ .It follows from this lower bound and the above upper bound, that we have the relativeposition is given precisely by w − w P w ′ , as required. (cid:3) Further from the above computations we can also understand the exact number of com-ponents in the intersection of each e P -orbit. Lemma 5.9.
Assume w satisfies equations (1) , (2) and (3) i precisely when i ∈ I w for somesubset I w ⊂ [ m − . Consider the subgroup S I w ⊂ S m generated by the simple reflections s i for i ∈ I w .Then Y w contains S m /S I w irreducible components.Proof. As we have seen above we have a map e P wI/I ∩ F l N → e P /I has image given by aHessenberg variety and over that is a recursive affine space bundle.Further Y ◦ w is smooth and it thus follows that the irreducible components coincide with theconnected components and these coincide with the connected components of the Hessenbergvariety.Again recall we have several maps e P /I → SL m /B and the image of the Hessenberg varietyis contained in a Hessenberg variety corresponding to a regular semisimple map on SL m .Note that the preimage under this map of each connected component is connected. Thisfollows by Lemma 5.7 and the proof of Lemma 5.6.Consider the parabolic P I ⊂ SL m generated by S I and the Borel subgroup B . It followsfrom the description of the Hessenberg variety H h as described in [2, 3] that H Wh is a subsetof S m P I w /B and the intersection of P I w /B with the Hessenberg variety is connected as it canbe described as a product of Hessenberg varieties satisfying the conditions of connectedness.It follows from this that the number of connected components are precisely given by thenumber of connected components of S m P I w /B . This is easily seen to be given by S m /S I w .This gives a multinomial coefficient which describes the number of irreducible componentsof Y w . (cid:3) We end this section by giving a description of every component as a component of thefundamental box after we apply the action of the F i as described in Section 4. We furtheridentify the Y w for general w in terms of this description. Theorem 5.10. (1)
Given any A w satisfying equations (1) and (2) (i.e. w ∈ w λ · R T λ ),we have Y w = [ σ ∈ S m m − Y i =1 F d i + d i +1 + ... + d m − σ ( i ) ( Y w ′ ) here d j = Diff j ( w − ) − lch j ( P ( w − w λ )) for j ∈ [ m − and w ′ ∈ F satisfies w = w ( k ) φ k ( w ( k ) φ k ( · · · w ( k ε ) φ k ε ( w ′ ) · · · )) for { k , . . . , k ε } = { d , d , . . . , ( m − d m − } as a multi-set. (2) All irreducible components of F l N are given without repetition by F c ( Y w ) for w ∈ F , c ∈ N m and min { c , . . . , c m } = 0 . Proof.
To begin consider the action on the points f W ∩ F l N . The points of f W are sent tothemselves under the action of F i and thus it follows that the action preserves the set ofelements of f W that satisfy equation (1) .For an m -tuple c recall that we define F c = F c . . . F c m m . We consider c weakly decreasing c ≥ c ≥ . . . ≥ c m .We begin by first claiming that F k ( w ) = w ( k ) φ k ( w ) where k is the m -tuple given by k ’s followed by m − k ’s. In fact, by direct computation, s − k f ′− . . . f ′− k = Ds − k [1+ m, m, . . . , k + m, k, . . . , m, m, . . . , k +2 m, k + m, . . . , m, . . . , n ]= D [1+ m − k, m − k, . . . , m, , . . . , m − k, m − k, . . . , m, m, . . . , m − k, . . . , n ]= Dφ − k ( w ( k ) ) . where D = diag { , . . . , , h − , . . . , h − k } . Hence f ′− . . . f ′− k = φ k ( D ) w ( k ) s k and the claimfollows.Now we prove that F c ( w P IwI/I ) ⊂ e P F c ( w ) I/I for c weakly decreasing and w satisfying (1) and (2) . Note that w P IwI/I is an open subset of e P wI/I and thus it will follow that thecomponent X of Y w containing w satisfies that F c ( Y w ) ⊂ Y F c ( w ) .To check the statement note that it is enough to prove ad ( f ′ ) c ( U α ) ⊂ e P for the root spaces U α such that w P ( α ) is positive and w − ( α ) is negative. To check this we just note that aroot e i − e j + kδ can satisfy this only if i < j , where i ≡ i (mod m) and i ∈ [ m ] . Note thatad ( f ′ ) c ( U α ) ⊂ e P if and only if c i + (cid:4) im (cid:5) − c j − (cid:4) jm (cid:5) ≥ . Note that as U α ⊂ e P , we have (cid:4) im (cid:5) − (cid:4) jm (cid:5) ≥ , thus the result follows as i < j ⇒ c i ≥ c j as c is weakly decreasing.Consider the action of S m ֒ → W P the diagonal group. This acts on N by sending it toa similar element, but with the regular semisimple h exchanged by the action of x ∈ S m on it x h . Note that these two are part of the family F l N and so there is a monodromyaction on the components on this space. Note that the e P -orbits are preserve, so there isan action on Y w . Note that the action on the fix points W P w ֒ → e P wI/I is given by leftmultiplication. Note that the component of Y w are determined by the intersection with thepoints S m w ֒ → W p w . Thus we get all the components of Y w are related by this actionUsing this we can check that for σ ∈ S m σ ( F c ( Y w )) = F σ ( c ) ( Y w ) . Thus for c weaklydecreasing, we have ∪ σ ∈ S m F σ ( c ) ( Y w ) = Y F c ( w ) as we have an inclusion and both have thesame number of components. From Corollary 3.12, we have the iterative expression w = w ( k ) φ k ( w ( k ) φ k ( · · · w ( k ε ) φ k ε ( w ′ ) · · · )) for any w ∈ w λ · R T λ , hence the first result is proved.Since F l N = S w ∈ w λ · R Tλ Y w , we arrive at the second result by putting all irreducible com-ponents of Y w ’s in the first result together. (cid:3) Remark . By Corollary 3.7, we know φ m ( w ) ∈ F if w ∈ F . Hence it follows from theproof of Theorem 5.10 that F ( Y w ) = Y φ m ( w ) (where = m in the vector of all 1’s) is rreducible when w ∈ F . This provides an alternative explanation of the condition c ∈ N m and min { c , . . . , c m } = 0 in Theorem 5.10 (2).We end with a lemma to understand the relative position of the components with theabove description. Lemma 5.12. r ( F k ( Y w ′ ) , F c ( Y w )) = φ k ( r ( Y w ′ , F c − k ( Y w ))) for any w ′ , w ∈ F .Proof. We will just prove that r ( F i ( x ) , F i ( y )) = φ ( r ( x, y )) , the result then follows, as therelative position of irreducible subsets is just the generic relative position of pairs of points.Note that F i is given by multiplication by ( f ′ i ) − followed by shifting the lattices by one.Note that this shifting can by understood as the following transformation gI gs − I , where s is the affine permutation introduced before Lemma 2.1.This is indeed well defined, because sIs − = I . Thus we have if x = gI , y = g ′ I , then r ( x, y ) = r ( I, g − g ′ I ) , where we have g − g ′ I ∈ Ir ( I, g − g ′ I ) I . Note than r ( F i ( x ) , F i ( y )) = r ( I, sg − g ′ s − I ) and sg − g ′ s − I ∈ Isr ( I, g − g ′ I ) s − I , but note that sws − = φ ( w ) , hencethe result follows as required. (cid:3) The case of n = 2 The theorems about the Knuth equivalence classes in the Section 2 do not work for n = 2 ,so we discuss the n = 2 case separately in this section, where both geometry and combina-torics can be explicitly computed.For n = 2 , e S is generated by s = [2 , and s = [0 , with relations s = s = id .There are two partitions (2) and (1 , of size . The two-sided cells C (2) = { [1 , } and C (1 , = e S \ C (2) . Explicit computation of affine matrix ball construction gives Φ (cid:0) [2 k + 2 , − k ] = s ( s s ) k (cid:1) = , , (cid:18) − kk (cid:19)! , k ≥ (cid:0) [ − − k, k ] = ( s s ) k +1 (cid:1) = , , (cid:18) − kk (cid:19)! , k ≥ (cid:0) [2 k + 1 , − k ] = ( s s ) k (cid:1) = , , (cid:18) − kk (cid:19)! , k ≥ (cid:0) [ − k, k ] = s ( s s ) k (cid:1) = , , (cid:18) − kk (cid:19)! , k ≥ . Now we consider the affine Springer fibers of two types. When N = (cid:18) t (cid:19) , F l N = { I/I } ,a singleton, and r ( I/I, I/I ) = id ∈ C (2) .When N = (cid:18) t − t (cid:19) , F l N is an infinite chain of P ’s. Explicitly, it consists of irreduciblecomponents { C k = F k ( C ) } k ∈ Z , where C = G ( O ) /I = { V. ∈ F l | V = Oh e , e i} ≃ G/B and for k ∈ Z , C k = { V. ∈ F l | V = Oh t − k e , t k e i} ,C k +1 = { V. ∈ F l | V = Oh t − k − e , t k e i} . oreover, each C k intersects only with C k − and C k +1 at precisely one flag.The entire affine Springer fiber is the union of Y w = G ( O ) wI/I ∩ F l N for w satisfying w − (1) < w − (2) . Y id = C and Y w = C k ⊔ C − k for w = [1 − k, k ] − and k ≥ . Then r ( C , C k ) ≤ r ( Y id , Y w ) ≤ s ∗ s w = s w for w = [1 − k, k ] − . And by the explicit shapeof C k ’s above, this upper bound can be reached by the relative position of certain coordinateflags in two components respectively. Hence r ( C , C k ) = s s s s . . . where there are k + 1 terms. Similarly r ( C , C k ) = s s s s . . . where there are k terms.Since Λ = (cid:28) F F − = (cid:18) t − t (cid:19)(cid:29) and F F − ( C k ) = C k +2 , we have established the follow-ing commutative diagrams where θ and Θ are bijections: Irr( F l N ) / Λ Irr( F l N ) × Λ Irr( F l N ) e S T ((1 , (1 , θ pr i r Θ pr i Ψ where pr i , i = 1 , are the natural projection maps onto the first and second component andfor all k ∈ Z , θ ( C k ) = 12 , θ ( C k +1 ) = 21 and r ( C , C k ) = ( s s ) | k | s , Θ( C , C k ) = , , (cid:18) − kk (cid:19)! ,r ( C , C k +1 ) = (cid:26) ( s s ) k +1 , k ≥ s s ) − k , k ≤ − , Θ( C , C k +1 ) = , , (cid:18) − kk (cid:19)! ,r ( C , C k ) = (cid:26) ( s s ) k , k ≥ s s ) − k +1 , k ≤ , Θ( C , C k +1 ) = , , (cid:18) − kk (cid:19)! ,r ( C , C k +1 ) = ( s s ) | k | s , Θ( C , C k ) = , , (cid:18) − kk (cid:19)! . The main theorem
In this final section, we explicitly compute the image of the relative positions betweenirreducible components of F l N (obtained in Theorem 5.8) under the affine matrix ball con-struction. Then we conclude with the bijection from pairs of irreducible components modulocommon translations to the triples ( P, Q, ρ ) of rectangular shape and obtain an analogue ofSteinberg and van Leeuven’s result. The case of n = 2 is explicitly computed in the previoussection, so we focus on n ≥ here. Recall from Theorem 5.10 that all irreducible componentsof F l N are given without repetition by F c ( Y w ) for w ∈ F , c ∈ N m and min { c , . . . , c m } = 0 . Proposition 7.1.
For w ′ , w ′′ ∈ F , c ∈ N m and min { c , . . . , c m } = 0 , we have Φ ( r ( Y w ′ , F c ( Y w ′′ ))) = (cid:16) ( w ′ ) − ( T λ ) , ( w ′′ ) − ( T λ ) + | c | , ρ (cid:17) , here | c | = c + . . . + c m ,ρ = ˜ ρ + s (cid:16) ( w ′ ) − ( T λ ) (cid:17) − s (cid:16) ( w ′′ ) − ( T λ ) + | c | (cid:17) , ˜ ρ = − ˜ ρ ( w λ w ′ ) + ˜ ρ ( w λ w ′′ ) − (cid:18)(cid:22) | c | n (cid:23) l + δ | c | (cid:16) ( w ′′ ) − (cid:17)(cid:19) + c dom , and the function δ is defined to be δ αi ( u ) = l − X j =0 [ n − α +1 ,n ] ( u ( i + jm )) , α ∈ [0 , n − , i ∈ [ m ] . Proof.
Let c ′ ≥ c ′ ≥ . . . ≥ c ′ m be the weakly decreasing rearrangement of c , . . . , c m , and d i = c ′ i − c ′ i +1 for i ∈ [ m − and denote { k , . . . , k ε } = { d , d , . . . , ( m − d m − } . So k + . . . + k ε = | c | . From Theorem 5.8, we know r ( Y w ′ , F c ( Y w ′′ )) = ( w ′ ) − w λ w where w ∈ w λ · R w λ such that F c ( Y w ′′ ) ⊂ Y w . And from the proof of Proposition 3.9, we have ( w ′ ) − w λ w ∼ LKC w λ w , ( w ′ ) − w λ w and ( w ′ ) − w λ are in the same right cell, so we know by Lemma 2.5 and Theorem2.15: P (( w ′ ) − w λ w ) = P (( w ′ ) − w λ ) = ( w ′ ) − ( T λ ) ,Q (( w ′ ) − w λ w ) = Q ( w λ w ) = w − ( T λ ) ,ρ (( w ′ ) − w λ w ) = ρ (( w ′ ) − w λ ) + ρ ( w λ w ) . Since w = w ( k ) φ k ( w ( k ) φ k ( · · · w ( k ε ) φ k ε ( w ′′ ) · · · )) , Q (( w ′ ) − w λ w ) = w − ( T λ ) = ( w ′′ ) − ( T λ ) + k + . . . + k s = ( w ′′ ) − ( T λ ) + | c | . From Proposition 2.10, it suffices to show the following claim: ˜ ρ ( w − w λ ) − ˜ ρ (( w ′′ ) − w λ ) = (cid:18)(cid:22) | c | n (cid:23) l + δ | c | (cid:16) ( w ′′ ) − (cid:17)(cid:19) + ( − c ) dom . (2)Denote u ( ε ) = w ′′ and u ( r ) = w ( k r +1 ) φ k r +1 ( u ( r +1) ) for r ∈ [0 , ε − .Since u ( r ) ∈ w λ · R w λ , we have ρ i (( u ( r ) ) − w λ ) = P l − j =0 l ( u ( r ) ) − ( i + jn ) n m − l for any r ∈ [0 , ε ] and i ∈ [ m ] . Explicitly, ( u ( r ) ) − equals [ ( u ( r +1) ) − (1 + ( l − m ) − n + k, . . . , ( u ( r +1) ) − ( k + ( l − m ) − n + k, ( u ( r +1) ) − ( k + 1) + k, . . . , ( u ( r +1) ) − ( m ) + k, ( u ( r +1) ) − (1) + k, . . . , ( u ( r +1) ) − ( k ) + k, ( u ( r +1) ) − ( k + 1 + m ) + k, . . . , ( u ( r +1) ) − (2 m ) + k,. . . ( u ( r +1) ) − (1 + ( l − m ) + k, . . . , ( u ( r +1) ) − ( k + ( l − m ) + k, ( u ( r +1) ) − ( k + 1 + ( l − m ) + k, . . . , ( u ( r +1) ) − ( n ) + k ] . Therefore, ρ i (( u ( r ) ) − w λ ) = ρ i (( u ( r +1) ) − w λ ) − ( k r +1 ) i + δ k r +1 (cid:16) ( u ( r +1) ) − (cid:17) . umming over r ∈ [0 , ε − , we obtain: ρ ( w − w λ ) = ρ (( w ′′ ) − w λ ) − ε X r =1 k r + ε X r =1 δ k r (cid:16) ( u ( r ) ) − (cid:17) = ρ (( w ′′ ) − w λ ) − ε X r =1 k r + δ | c | (cid:16) ( w ′′ ) − (cid:17) + (cid:22) | c | n (cid:23) l . (3)Note that | c | in the δ function is taken to be in [0 , n − .Now we consider the entries of the symmetrized offset constant vectors. (cid:16) s (cid:16) w − ( T λ ) + | c | (cid:17)(cid:17) i = l − X j =0 (cid:18)(cid:24) w − ( i + jm ) n (cid:25) − (cid:24) w − (1 + jm ) n (cid:25)(cid:19) − ( d + . . . + d i − ) , (4) (cid:16) s (cid:16) ( w ′′ ) − ( T λ ) + | c | (cid:17)(cid:17) i = l − X j =0 (cid:18)(cid:24) ( w ′′ ) − ( i + jm ) n (cid:25) − (cid:24) ( w ′′ ) − (1 + jm ) n (cid:25)(cid:19) . (5)Also from the explicit form of ( u ( r ) ) − , we have: l − X j =0 (cid:24) ( u ( r ) ) − ( i + jm ) n (cid:25) = l − X j =0 (cid:24) ( u ( r +1) ) − ( i + jm ) n (cid:25) − ( k r +1 ) i + δ k r +1 i (cid:16) ( u ( r +1) ) − (cid:17) . So l − X j =0 (cid:18)(cid:24) ( u ( r ) ) − ( i + jm ) n (cid:25) − (cid:24) ( u ( r ) ) − (1 + jm ) n (cid:25)(cid:19) = l − X j =0 (cid:18)(cid:24) ( u ( r +1) ) − ( i + jm ) n (cid:25) − (cid:24) ( u ( r +1) ) − (1 + jm ) n (cid:25)(cid:19) − ( k r +1 ) i + δ k r +1 i (cid:16) ( u ( r +1) ) − (cid:17) + 1 − δ k r +1 (cid:16) ( u ( r +1) ) − (cid:17) . Summing over r ∈ [0 , ε − again and rearranging terms, we get l − X j =0 (cid:18)(cid:24) w − ( i + jm ) n (cid:25) − (cid:24) w − (1+ jm ) n (cid:25)(cid:19) − l − X j =0 (cid:18)(cid:24) ( w ′′ ) − ( i + jm ) n (cid:25) − (cid:24) ( w ′′ ) − (1+ jm ) n (cid:25)(cid:19) = − ε X r =1 ( k r ) i + δ k + ... + k ε i (cid:16) ( w ′′ ) − (cid:17) + (cid:22) k + . . . + k ε n (cid:23) l + ε − δ k + ... + k ε (cid:16) ( w ′′ ) − (cid:17) . (6)Equations (4), (5), (6) together give s ( P ( w − w λ )) − s ( P (( w ′′ ) − w λ ))= − ε X r =1 k r + δ k + ... + k ε (cid:16) ( w ′′ ) − (cid:17) + (cid:16) s − δ k + ... + k ε (cid:16) ( w ′′ ) − (cid:17)(cid:17) − d d + d . . .d + . . . + d m − . (7) y definition of the d i ’s, we have d d + d . . .d + . . . + d m − = c ′ − c ′ c ′ − c ′ c ′ − c ′ . . .c ′ − c ′ m = c ′ + ( − c ) dom = s + ( − c ) dom . (8)Plugging (8) into (7) and subtracting (7) from (3), we arrived at the claim (2). (cid:3) We illustrate Proposition 7.1 using the following example.
Example 7.2.
Let n = 6 , λ = (2 , , and take C = Y id and C = F i F j ( Y w ′′ ) where i, j ∈ [3] , i = j and ( w ′′ ) − = [0 , , , , , ∈ F − (which can be read from Figure 3 in theappendix). Then we know { k , . . . , k } = { , , , , } and C ⊂ Y w where w = w (2) φ ( w (2) φ ( w (1) φ ( w (1) φ ( w (1) φ ( w ′′ ))))) . Direct computation gives w − = [ − , , , − , , . And we know x := r ( C , C ) = w λ w =[12 , , − , , − , and P ( x ) = 1 42 53 6 , Q ( x ) = 1 42 63 5 , ρ ( x ) = − . Now we use the formula in Proposition 7.1 to reproduce Φ( x ) . P ( x ) = ( w ′ ) − ( T λ ) = T λ ,Q ( x ) = ( w ′′ ) − ( T λ ) + c + c + c = 3 61 52 4 + 7 = 1 42 63 5 ,ρ ( x ) = s − s − ˜ ρ ( w λ w ′ ) + ˜ ρ ( w λ w ′′ ) − (cid:18)(cid:22) c + c + c n (cid:23) l + δ c + c + c (cid:16) ( w ′′ ) − (cid:17)(cid:19) + c dom = − − + − (2 + 1) + = − . Theorem 7.3.
Let w ′ , w ′′ ∈ F , c ≥ , . . . , c m ≥ , min { c , . . . , c m } = 0 , and k ∈ [0 , m − .Then, Φ (cid:0) r ( F k ( Y w ′ ) , F c ( Y w ′′ )) (cid:1) = (cid:16) ( w ′ ) − ( T λ ) + k, ( w ′′ ) − ( T λ ) + | c | , ρ (cid:17) , here ρ = ˜ ρ + s (cid:16) ( w ′ ) − ( T λ ) + k (cid:17) − s (cid:16) ( w ′′ ) − ( T λ ) + | c | (cid:17) , ˜ ρ = − ˜ ρ ( w λ w ′ ) + ˜ ρ ( w λ w ′′ ) − (cid:18)(cid:22) | c | n (cid:23) l + δ | c | (cid:16) ( w ′′ ) − (cid:17) − δ k (( w ′ ) − ) (cid:19) + ( c − k ) dom , and the δ function is defined to be δ αi ( u ) = l − X j =0 [ n − α +1 ,n ] ( u ( i + jm )) , α ∈ [0 , n − , i ∈ [ m ] . Proof.
Denote x := r ( F k ( Y w ′ ) , F c ( Y w ′′ )) and z := r ( Y w ′ , F c − k ( Y w ′′ )) . From Lemma 5.12,we know that x = φ k ( z ) .Case 1: min { c , . . . , c k } ≥ .In this case we could apply Proposition 7.1 directly. Take c ′ ≥ c ′ ≥ . . . ≥ c ′ m be the weaklydecreasing rearrangement of c − , . . . , c k − , c k +1 , . . . , c m , and d i = c ′ i − c ′ i +1 for i ∈ [ m − and denote { k , . . . , k ε } = { d , d , . . . , ( m − d m − } . Note here k + . . . + k ε = c + . . . c m − k. Then F c − k ( Y w ′′ ) ⊂ Y w where w = w ( k ) φ k ( w ( k ) φ k ( · · · w ( k ε ) φ k ε ( w ′′ ) · · · )) .Hence by Lemma 2.11, we have: P ( x ) = P ( z ) + k = ( w ′ ) − ( T λ ) + k,Q ( x ) = Q ( z ) + k = ( w ′′ ) − ( T λ ) + ( c −
1) + . . . + ( c k −
1) + c k +1 + . . . + c m + k = ( w ′′ ) − ( T λ ) + | c | , and ρ ( x ) = ρ ( z ) + δ k (cid:16) ( w ′ ) − (cid:17) − δ k (cid:16) ( w ′′ ) − + k + . . . + k s (cid:17) = s (cid:16) ( w ′ ) − ( T λ ) (cid:17) − s (cid:16) ( w ′′ ) − ( T λ ) + | c | − k (cid:17) − ˜ ρ ( w λ w ′ ) + ˜ ρ ( w λ w ′′ ) − (cid:18)(cid:22) | c | − kn (cid:23) l + δ | c |− k (cid:16) ( w ′′ ) − (cid:17)(cid:19) + ( c − k ) dom + δ k (cid:16) ( w ′ ) − (cid:17) − δ k (cid:16) ( w ′′ ) − + | c | − k (cid:17) . (9)Since s i (cid:16) ( w ′ ) − ( T λ ) + k (cid:17) = s i ( P ( w ( k ) φ k ( w ′ ))) − ( − k ) , we have s (cid:16) ( w ′ ) − ( T λ ) (cid:17) = s (cid:16) ( w ′ ) − ( T λ ) + k (cid:17) + δ k (cid:16) ( w ′ ) ( − (cid:17) − δ k (cid:16) ( w ′ ) ( − (cid:17) . (10)Similarly, there is s (cid:16) ( w ′′ ) − ( T λ ) + | c | − k (cid:17) = s (cid:16) ( w ′′ ) − ( T λ ) + | c | (cid:17) + δ k (cid:16) ( w ′′ ) − + | c | − k (cid:17) − δ k (cid:16) ( w ′′ ) − + | c | − k (cid:17) . (11) quations (9), (10), (11) together give ˜ ρ ( x ) = δ k (cid:16) ( w ′ ) ( − (cid:17) − δ k (cid:16) ( w ′′ ) − + | c | − k (cid:17) − (cid:18)(cid:22) | c | − kn (cid:23) l + δ | c |− k (cid:16) ( w ′′ ) − (cid:17)(cid:19) − ˜ ρ ( w λ w ′ ) + ˜ ρ ( w λ w ′′ ) + ( c − k ) dom = − (cid:18)(cid:22) | c | n (cid:23) l + δ | c | (cid:16) ( w ′′ ) − (cid:17) − δ k (( w ′ ) − ) (cid:19) − ˜ ρ ( w λ w ′ ) + ˜ ρ ( w λ w ′′ ) + ( c − k ) dom . Case 2: min { c , . . . , c k } = 1 .In this case, we need to use Corollary 3.7 and write the component as F c − k ( Y w ′′ ) = F c + − k ( Y φ − m ( w ′′ ) ) . Now we could apply Proposition 7.1. Denote y = φ − m ( w ′′ ) . and take c ′ ≥ c ′ ≥ . . . ≥ c ′ m bethe weakly decreasing rearrangement of c , . . . , c k , c k +1 + 1 , . . . , c m + 1 , and d i = c ′ i − c ′ i +1 for i ∈ [ m − and { k , . . . , k ε } = { d , d , . . . , ( m − d m − } . Here k + . . . + k ε = | c | + m − k. Similarly, F c + − k ( Y y ) ⊂ Y w where w = w ( k ) φ k ( w ( k ) φ k ( · · · w ( k ε ) φ k ε ( y ) · · · )) . Again byLemma 2.11, we have P ( x ) = P ( z ) + k = ( w ′ ) − ( T λ ) + k,Q ( x ) = Q ( z ) + k = y − ( T λ ) + c + . . . + c k + ( c k +1 + 1) + . . . + ( c m + 1) + k = (cid:16) y − ( T λ ) + m (cid:17) + | c | = ( w ′′ ) − ( T λ ) + | c | , and ρ ( x ) = ρ ( z ) + δ k (cid:16) ( w ′ ) − (cid:17) − δ k (cid:16) y − + | c | + m − k (cid:17) = s (cid:16) ( w ′ ) − ( T λ ) (cid:17) − s (cid:16) y − ( T λ ) + | c | + m − k (cid:17) − ˜ ρ ( w λ w ′ ) + ˜ ρ ( w λ y ) − (cid:18)(cid:22) | c | + m − kn (cid:23) l + δ | c | + m − k (cid:16) y − (cid:17)(cid:19) + ( c + − k ) dom + δ k (cid:16) ( w ′ ) − (cid:17) − δ k (cid:16) y − + | c | + m − k (cid:17) . (12)Explicitly, y − = [ ( w ′′ ) − (1 + m ) − m, ( w ′′ ) − (2 + m ) − m, . . . , ( w ′′ ) − (2 m ) − m, ( w ′′ ) − (1 + 2 m ) − m,. . . , ( w ′′ ) − ( n ) − m, ( w ′′ ) − (1) + n − m, . . . , ( w ′′ ) − ( m ) + n − m (cid:3) . Hence there is s (cid:16) y − ( T λ ) + | c | + m − k (cid:17) = s (cid:16) ( w ′′ ) − ( T λ ) + | c | − k (cid:17) = s (cid:16) ( w ′′ ) − ( T λ ) + | c | (cid:17) + δ k (cid:16) ( w ′′ ) − + | c | − k (cid:17) − δ k (cid:16) ( w ′′ ) − + | c | − k (cid:17) . (13) ombining equations (10), (12), (13), we obtain ˜ ρ ( x ) = − ˜ ρ ( w λ w ′ ) + ˜ ρ ( w λ y ) + δ k (cid:16) ( w ′ ) − (cid:17) − δ k (cid:16) ( w ′′ ) − + | c | − k (cid:17) − (cid:18)(cid:22) | c | + m − kn (cid:23) l + δ | c | + m − k (cid:16) ( w ′′ ) − − m (cid:17)(cid:19) + ( c + − k ) dom = − ˜ ρ ( w λ w ′ ) + ˜ ρ ( w λ y ) + δ k (cid:16) ( w ′ ) − (cid:17) + ( c + − k ) dom − (cid:18)(cid:22) | c | + mn (cid:23) l + δ | c | + m (cid:16) ( w ′′ ) − − m (cid:17)(cid:19) . (14)Finally, ˜ ρ ( w λ y ) = − ˜ ρ ( y − w λ ) = − − l + l − X j =0 (cid:24) ( w ′′ ) − (1 + jm ) − mn (cid:25)! = − ˜ ρ (( w ′′ ) − w λ ) − − l − X j =0 [1 ,m ] (cid:16) ( w ′′ ) − (1 + jm ) (cid:17)! . (15)Let α = | c | + m ∈ [0 , n − . Then − l − X j =0 [1 ,m ] (cid:16) ( w ′′ ) − (1 + jm ) (cid:17) + l − X j =0 [ n − α +1 ,n ] (cid:16) ( w ′′ ) − (1 + jm ) (cid:17) = l − X j =0 [ n − α + m +1 ,n ] (cid:16) ( w ′′ ) − (1 + jm ) (cid:17) , α ≥ m − [1 ,m − α ] (cid:16) ( w ′′ ) − (1 + jm ) (cid:17) , α < m = l − X j =0 [ n − α + m +1 ,n ] (cid:16) ( w ′′ ) − (1 + jm ) (cid:17) , α ≥ m − [ m − α +1 ,n ] (cid:16) ( w ′′ ) − (1 + jm ) (cid:17) , α < m (16)Combining (14), (15) and (16) gives the same expression of ρ ( x ) as in Case 1. (cid:3) Example 7.4.
Let n = 6 , λ = (2 , , and C = F F ( Y w ′ ) , C = F F ( Y w ′′ ) where ( w ′ ) − =[ − , , , , , and ( w ′′ ) − = [0 , , , , , . It can be checked from Figure 3 in the appendixthat w ′ , w ′′ ∈ F . Then y − = φ − m (( w ′′ ) − ) = φ − ([0 , , , , , , , , , , and F − F F ( Y w ′′ ) = F F ( Y y ) ⊂ Y w where w = w (2) φ ( w (2) φ ( w (1) φ ( w (1) φ ( w (1) φ ( w (1) φ ( w ′′ )))))) = [ − , , , − , , − . So x := r ( C , C ) = φ (( w ′ ) − w λ w ) = [15 , , − , , − , . Direct calculation by affine matrix ball construction gives Φ( x ) = Φ([15 , , − , , − , , , − . ow we use Theorem 7.3 to calculate the same data. P ( x ) = ( w ′ ) − ( T λ ) + k = 3 52 61 4 + 2 = 1 52 43 6 ,Q ( x ) = ( w ′′ ) − ( T λ ) + c + c + c = 3 61 52 4 + 7 = 1 42 63 5 ,ρ ( x ) = s − s − ˜ ρ ( w λ w ′ ) + ˜ ρ ( w λ w ′′ )+ (cid:16) δ k (cid:16) ( w ′ ) − (cid:17)(cid:17) − (cid:18)(cid:22) c + c + c n (cid:23) l + δ c + c + c (cid:16) ( w ′′ ) − (cid:17)(cid:19) + ( c − k ) dom = − − + + − (2 + 1) + − = − . The main theorem of the paper is as follows:
Theorem 7.5.
For λ = ( l m ) , we have the following commutative diagrams: Irr( F l N ) / Λ Irr( F l N ) × Λ Irr( F l N ) f S n T ( λ ) Ω λθ pr i r Θ pr i Ψ (17) where θ ( F c ( Y w )) = φ mα ( w − )( T λ ) + β, here | c | = mα + β, α ∈ Z , β ∈ [0 , m − , Ω λ = { ( P, Q, ρ ) ∈ Ω | P, Q ∈ T ( λ ) , ρ ∈ Z m } , Θ (cid:0) F k ( Y w ′ ) , F c ( Y w ′′ ) (cid:1) = (cid:16) ( w ′ ) − ( T λ ) + k, ( w ′′ ) − ( T λ ) + | c | , ρ (cid:17) for w ′ , w ′′ ∈ F ,ρ = s (cid:16) ( w ′ ) − ( T λ ) + k (cid:17) − s (cid:16) ( w ′′ ) − ( T λ ) + | c | (cid:17) + − ˜ ρ ( w λ w ′ ) + ˜ ρ ( w λ w ′′ ) − (cid:18)(cid:22) | c | n (cid:23) l + δ | c | (cid:16) ( w ′′ ) − (cid:17) − δ k (( w ′ ) − ) (cid:19) + ( c − k ) rev ,δ αi ( u ) = l − X j =0 [ n − α +1 ,n ] ( u ( i + jm )) , α ∈ [0 , n − , i ∈ [ m ] . nd pr i , i = 1 , are the natural projection maps onto the first and second component.Moreover, these maps satisfy: (1) The relative position r maps onto the two-sided cell C λ ; (2) θ and Θ are bijections; (3) Given any C ∈ Irr( F l N ) / Λ , then r ( pr − ( C )) is a right cell in C λ , and r ( pr − ( C )) isa left cell in C λ .Proof. For any x ∈ C λ , we use Proposition 3.9 and consider the right cell it is contained in.Then x = φ k (( w ′ ) − w w ) for some k ∈ [0 , m − , w ′ ∈ F = w λ · RKC w λ and w ∈ w λ · R w λ . Take C to be any component containing in Y w and by Lemma 5.12 we have r ( F k ( Y w ′ ) , F k ( C )) = x , therefore proving the first claim.From the formulas in Theorem 7.5 and Corollary 3.6, we know the only factor that affectsinjectivity of r is ( c − k ) dom . Lemma 5.9 and Proposition 3.13 show that Θ is a bijection.Finally, the expression of the insertion (resp. recording) tabloid in the image of Θ onlydepends on the first (resp. second) component, therefore r ( pr − ( C )) is a right cell and r ( pr − ( C )) is a left cell. (cid:3) Remark . From Lemma 5.9 and Proposition 3.13, the numbers of components in the ˜ P -orbit and the fibers of Ψ are the same (both possess a Weyl group symmetry). Hence there aremany ways to define the bijection Θ to make the diagram commute. When ( l m ) = (1 n ) , thedefinition of ρ presented in Theorem 7.5 differs by a rotation with the one in [25]. Whetherthere is a preferred choice reduces to the question of the geometric meaning of the weightvector ρ . Corollary 7.7.
When λ = ( l m ) , the relative position map induces the following bijection: r : Irr( F l N ) × ˜Λ Irr( F l N ) → C λ , where ˜Λ = Λ ⋊ S m . Explicitly, let C = F c ′ ( Y w ′ ) , C = F c ( Y w ) , D = F d ′ ( Y u ′ ) , D = F d ( Y u ) where c ′ , c, d ′ , d ∈ Z m and w ′ , w, u ′ , u ∈ F . Then ( C , C ) ∼ ( D , D ) iff the following threecriterions hold: (1) φ | c ′ | ( w ′ ) = φ | d ′ | ( u ′ ) , φ | c | ( w ) = φ | d | ( u ) ; (2) m | ( | c ′ | − | d ′ | ) , m | ( | c | − | d | ); (3) there exists σ ∈ S m , such that σ ( c − c ′ ) − ( d − d ′ ) is a constant vector.Proof. Let | c ′ | = mα ′ + β ′ , | c | = mα + β, | d ′ | = mγ ′ + η ′ , | d | = mγ + η where α ′ , α, γ ′ , γ ∈ Z and β ′ , β, η ′ , η ∈ [0 , m − . Then x := r ( C , C ) = r (cid:16) F β ′ ( Y φ mα ′ ( w ′ ) ) , F c − c ′ + α ′ + β ′ ( Y w ) (cid:17) = φ β ′ (cid:16) r (cid:16) Y φ mα ′ ( w ′ ) , F c − c ′ + α ′ ( Y w ) (cid:17)(cid:17) . Similarly, y := r ( D , D ) = r (cid:16) F η ′ ( Y φ mγ ′ ( u ′ ) ) , F d − d ′ + γ ′ + η ′ ( Y u ) (cid:17) = φ η ′ (cid:16) r (cid:16) Y φ mγ ′ ( u ′ ) , F d − d ′ + γ ′ ( Y u ) (cid:17)(cid:17) . Since x = y , we know β ′ = η ′ . Applying Proposition 7.1 and P ( φ − β ′ ( x )) = P ( φ − η ′ ( y )) ,we have φ mα ′ ( w ′ ) = φ mγ ′ ( u ′ ) . And Q ( φ − β ′ ( x )) = Q ( φ − η ′ ( y )) indicates w − ( T λ ) + | c | = u − ( T λ ) + | d | , which is equivalent to β = η and φ mα ( w ) = φ mγ ( u ) .Finally, we have ρ ( φ − β ′ ( x )) = ρ ( φ − η ′ ( y )) , which is equivalent to saying F c − c ′ + α ′ ( Y w ) and F d − d ′ + γ ′ ( Y u ) are contained in the same e P -orbit. But F c − c ′ + α ′ ( Y w ) = F c − c ′ + α ′ − α ( Y φ mα ( w ) ) , nd F d − d ′ + γ ′ ( Y u ) = F d − d ′ + γ ′ − γ ( Y φ mγ ( u ) ) . So there exists σ ∈ S m , such that σ ( c − c ′ + α ′ − α ) = d − d ′ + γ ′ − γ . (cid:3) A natural conjecture is the following:
Conjecture 7.8.
For every partition λ of n , there exists bijective maps θ λ and Θ λ that makethe following two diagrams commutative: Irr( F l N ( λ ) ) / Λ N ( λ ) Irr( F l N ( λ ) ) × Λ N ( λ ) Irr( F l N ( λ ) ) f S n T ( λ ) Ω λθ λ pr i r Θ λ pr i Ψ (18) where N ( λ ) ∈ g ( K ) is a generic lift of a nilpotent element in g of type λ and pr i , i = 1 , are the natural projection maps onto the first and second component. Appendix A. Diagrams of left Knuth classes containing w λ In this appendix, we show graphs of
LKC w λ in case λ = (2 , , (3 , and (2 , , , wherethe edges corresponds to left Knuth moves. [2 , , , , , ,
2] [2 , , , s · s · Figure 1.
LKC w λ when λ = (2 , , , , , , , , , , ,
5] [2 , , , , ,
4] [2 , , , , , , , , , ,
4] [3 , , , , ,
5] [2 , , , , , , , , , ,
3] [3 , , , , ,
5] [2 , − , , , , s · s · s · s · s · s · s · s · s · s · s · s · Figure 2.
LKC w λ when λ = (3 , , , , , , , , , , ,
3] [3 , , , , , , , , , ,
2] [5 , , , , ,
3] [4 , , , , ,
3] [3 , , − , , ,
4] [3 , , , , , , , , , ,
2] [5 , , , , ,
2] [5 , , , , ,
3] [4 , , − , , ,
3] [3 , , − , , ,
4] [4 , , , , , , , , , ,
1] [5 , , , , ,
2] [6 , , − , , ,
3] [5 , , − , , ,
3] [4 , , − , , ,
3] [4 , , , , , , , , , ,
1] [6 , , − , , ,
2] [6 , , − , , ,
3] [5 , , − , , ,
3] [4 , , − , , , , , − , , ,
1] [4 , , , , ,
1] [5 , , − , , , , , − , , ,
0] [5 , , − , , , s · s · s · s · s · s · s · s · s · s · s · s · s · s · s · s · s · s · s · s · s · s · s · s · s · s · s · s · s · s · s · s · s · s · s · s · s · s · s · s · s · s · F i g u r e . L K C w λ w h e n λ = ( , , ) eferences [1] Roman Bezrukavnikov, Pablo Boixeda Alvarez, Peng Shan, and Eric Vasserot. On the center of thesmall quantum group. upcoming work.[2] Pablo Boixeda Alvarez. Fix points and components of equivalued affine Springer fibers. arXiv preprintarXiv:1910.04780 , 2019.[3] Pablo Boixeda Alvarez. Affine Springer fibers and the representation theory of small quantum groupsand related algebras . PhD thesis, Massachusetts Institute of Technology, 2020.[4] Michael Chmutov, Joel Brewster Lewis, and Pavlo Pylyavskyy. Monodromy in Kazhdan-Lusztig cellsin affine type A. arXiv preprint arXiv:1706.00471 , 2017.[5] Michael Chmutov, Pavlo Pylyavskyy, and Elena Yudovina. Matrix-ball construction of affine Robinson-Schensted correspondence.
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E-mail address : Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA02139, USA
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