Regular strata and moduli spaces of irregular singular connections

Transcription

1 Regular sraa and moduli spaces of irregular singular connecions Daniel S. Sage In recen years, here has been exensive ineres in meromorphic connecions on curves due o heir role as Langlands parameers in he geomeric Langlands correspondence. In paricular, connecions wih irregular singulariies are he geomeric analogue of Galois represenaions wih wild ramificaion. The classical approach o sudying he local behavior of irregular singular meromorphic connecions on curves depends on he leading erm of he connecion marix being well-behaved. Le V be a rivializable vecor bundle over P 1 endowed wih a meromorphic (auomaically fla) connecion. Upon fixing a local parameer z a a singular poin y and a local rivializaion, one can express he connecion near y as d + (M r r + M 1 r 1 r +... ) d, (1) wih M i gl n (C), M r 0 and r 0. From a more geomeric poin of view, seing F = C(()), his formula defines he induced connecion ˆ y on he formal puncured disk Spec(F ). When M r is well-behaved, his leading erm conains imporan informaion abou he connecion. As a firs example, if M r is nonnilpoen, hen he expansion of a y wih respec o any local rivializaion mus begin in degree r or below. Moreover, if ˆ y is irregular, r is he slope of he connecion a y. (The slope is an invarian inroduced by Kaz [6] ha gives one measure of how singular a connecion is a a given poin.) Much more can be said in he irregular singular nonresonan case when r > 0 and M r is regular semisimple. We assume ha r > 0 so ha we are in he irregular singular case. In his case, asympoic analysis [9] guaranees ha can be diagonalized a y by an appropriae gauge change so ha = d + (D r r + D 1 r 1 r +... D 0 ) d, wih each D i diagonal. The The auhor was parially suppored by NSF gran DMS and Simons Foundaion Collaboraion Gran

2 2 Daniel S. Sage diagonal 1-form here is called a formal ype of ˆ y. When all of he irregular singulariies on a meromorphic connecion on P 1 are of his form, Boalch has shown how o consruc well-behaved moduli spaces of such connecions; he has furher realized he isomonodromy equaions as an inegrable sysem on an appropriae moduli space [1]. However, many ineresing connecions do no have regular semisimple leading erms. Consider, for example, he generalized Airy connecions: ( ) ( ) ( ) 0 (s+1) d 0 1 d + s 0 = d + (s+1) d s d 1 0, (2) for s 0. Noe ha when s = 1, his is he usual Airy connecion wih he irregular singular poin a 0 insead of. Also, when s = 0, his is he GL 2 version of he Frenkel-Gross rigid fla G-bundle on P 1 [7]. For he generalized Airy connecions, he leading erm is nilpoen, and i is no longer he case ha one can read off he slope direcly from he leading erm. Indeed, he slope is s + 1 2, no s + 1. In a recen series of papers join wih Bremer [2, 3, 4, 5], we have generalized hese classical resuls o meromorphic connecions on curves (or even fla G-bundles for reducive G) whose leading erm is nilpoen. We have inroduced a new noion of he leading erm of a formal connecion hrough a sysemaic analysis of is behavior in erms of suiable filraions on he loop algebra. This heory has already proved useful in applicaions o he geomeric Langlands program [8]. In his paper, we will illusrae our heory in he case of rank 2 fla vecor bundles, where much of he Lie-heoreic complexiy is absen. In his case, up o GL 2 (F )-conjugacy, one need only consider wo filraions on gl 2 (F ), he degree filraion and he (sandard) Iwahori filraion. Le o = C[[]] be he ring of formal power series, and le ω = ( ). Then he Iwahori filraion is defined by ( ) r/2 o r/2 o i r = r/2 +1 o r/2 o. (3) Recalling ha he sandard Iwahori subgroup I GL 2 (o) consising of he inverible marices which are upper riangular modulo, one sees ha i := Lie(I) is jus i 0 ; moreover, i r = iω r = ω r i. A marix is homogeneous of degree 2s (resp. 2s+1) wih respec o he Iwahori filraion if i is in ( ) C s 0 0 C (resp. ( ) s 0 C s C s+1 0 ). In paricular, he marix of he generalized Airy connecion is Iwahori-homogeneous of degree 2s + 1. The groups GL 2 (o) and I are examples of parahoric subgroups. For any parahoric P, here is an associaed filraion p j of gl 2 (F ); his filraion saisfies p j+e P = p j for e P {1, 2}. For GL 2 (o), he filered subspaces are j gl 2 (o). For simpliciy, we will ake P o be I or GL 2 (o) in his paper. Noe e GL2(o) = 1 and e I = 2. I will be convenien o view any one-form ν Ω 1 (gl 2 (F )) as a coninuous C-linear funcional on (subspaces of) gl 2 (F ) via Y Res Tr Y ν. Any such funcional on p r can be represened as X d for some X p r. For our

3 Regular sraa and moduli spaces of connecions 3 sandard examples, a funcional β (p r /p r+1 ) can be wrien uniquely as β d for β homogeneous. A GL 2 -sraum is a riple (P, r, β) wih P GL 2 (F ) a parahoric subgroup, r a nonnegaive ineger, and β (p r /p r+1 ). The sraum is called fundamenal if β is nonnilpoen. A formal connecion ˆ conains (P, r, β) if ˆ = d + X d wih X p r and β is induced by X d. The following heorem shows ha fundamenal sraa provide he correc noion of he leading erm of a connecion. Theorem 1. Any formal connecion ˆ conains a fundamenal sraum (P, r, β) wih r/e P = slope( ˆ ); in paricular, he connecion is irregular singular if and only if r > 0. Moreover, if (P, r, β ) is any oher sraum conained in ˆ, hen r /e P r/e P wih equaliy if (P, r, β ) is fundamenal. The converse hold if ˆ is irregular singular. The heorem shows ha he classical slope of a connecion can also be defined in erms of he fundamenal sraa conained in i. For fla G-bundles, he analogous resul serves o define he slope [4]. Example 1. The connecion in (2) (wih he M i s in gl 2 (C)) conains he r d sraum (GL 2 (o), r, M r ); i is fundamenal if and only if M r is nonnilpoen, in which case he slope is r. If M r is upper riangular wih a nonzero diagonal enry, hen ˆ conains a fundamenal sraum of he form (I, 2r, β), where β is induced by he diagonal componen of M r r. Again, one sees ha he slope is 2r/2 = r. On he oher hand, if M r has a nonzero enry below he diagonal, hen ˆ conains a nonfundamenal sraum of he form (I, 2r + 1, β ). Example 2. The generalized Airy connecion wih parameer s conains he nonfundamenal sraum (GL 2 (o), s + 1, ( ) 0 (s+1) d 0 0 ). I also conains he (2s+1) d fundamenal sraum (I, 2s + 1, ω ), whence is slope is s In order o consruc well-behaved moduli spaces, we need a condiion on sraa ha is analogous o he nonresonance condiion for diagonalizable connecions. This is accomplished hrough he noion of a regular sraum. Le S GL 2 (F ) be a (no necessarily spli) maximal orus. Up o GL 2 (F )- conjugacy, here are wo disinc maximal ori: T (F ) and C((ω) (nonzero Lauren series in ω). For our sandard examples, we say ha (P, r, β) is S- regular if S is he cenralizer of β. (See [2, 5] for he general definiion.) r d Example 3. If M r is regular semisimple, hen he sraum (GL 2 (o), r, M r ) is Z(M r )(F )-regular. (2s+1) d Example 4. The sraum (I, 2s + 1, ω ) conained in he generalized Airy connecion is C((ω)) -regular. On he oher hand, if (P, r, β) is C((ω)) - regular, hen r/e P 1 2 Z \ Z. From now on, we assume ha S is T (F ) or C((ω)). Noe ha s = Lie(S) is (F ) and C((ω)) in hese wo cases, and boh are endowed wih an

4 4 Daniel S. Sage obvious filraion by powers of or ω. We call a connecion conaining an S-regular sraum S-oral. An S-oral connecion can be diagonalized ino s = Lie(S). Again, for simpliciy, we will only describe wha his means for S equal o T (F ) and C((ω)). For any r > 0 such ha s r conains a regular semisimple elemen of homogeneous degree r, one can define a quasiaffine variey A(S, r) s r d of S-formal ypes of deph r: A(T, r) = {D r r + + D 0 D i, D r regular} d and A(C((ω)), 2s + 1) = {p(ω 1 ) d p C[ω 1 ], deg(p) = 2s + 1}. We remark ha if we se P T (F ) = GL n (o) and P C((ω)) = I, hen an S-formal ype A y = X d of deph r gives rise o he S-regular sraum (P S, r, X d ). Theorem 2. If ˆ conains he S-regular sraum (P, r, β), hen ˆ is P 1 := 1+ p 1 -gauge equivalen o a unique connecion of he form d + A for A A(S, r) wih leading erm β d. Before discussing moduli spaces, we need o define he noion of a framable connecion. Suppose ha is a fla G-bundle on P 1. Upon fixing a global rivializaion φ, we can wrie = d + [ ], where [ ] is he marix of he connecion. Assume ha he formal connecion ˆ y a y has formal ype A y. We say ha g GL 2 (C) is a compaible framing for a y if g ˆ y conains he regular sraum deermined by A y. For example, if A y = (D r r + + D 0 ) d, hen g is a global gauge change such ha g ˆ y = d + (D r r + X r+1 ) d wih X gl 2 (o). The connecion is framable a y if here exiss a compaible framing. We now explain how moduli spaces of connecions can be defined for meromorphic connecions on P 1 such ha ˆ y is oral a each irregular singulariy. We also wan o allow for regular singular poins. If he residue of a regular singular connecion is nonresonan, in he sense ha he eigenvalues do no differ by a nonzero ineger, hen is formal isomorphism class is deermined by he adjoin orbi of he residue. Accordingly, our saring daa will consis of: A nonempy se {x i } P 1 of irregular singular poins; A = (A i ), a se of S i -formal ypes wih posiive dephs r i a he x i s; A se {y j } P 1 of regular singular poins disjoin from {x i }; A corresponding collecion C = (C j ) of nonresonan adjoin orbis. Le M(A, C) be he moduli space classifying meromorphic rank 2 connecions (V, ) on P 1 wih V rivializable such ha: has irregular singular poins a he x i s, regular singular poins a he y j s, and no oher singular poins; is framable and has formal ype A i a x i ; has residue a y j in C j. We will consruc his moduli space as he Hamilonian reducion of a produc over he singular poins of symplecic manifolds, each of which is endowed wih a Hamilonian acion of GL 2 (C). A a regular singular poin wih adjoin orbi C, he symplecic manifold is C viewed as he coadjoin

5 Regular sraa and moduli spaces of connecions 5 orbi C d.) The symplecic manifold a an irregular singular poin wih formal ype A will be denoed M A ; i is called an exended orbi. To define i, le O A be he P S -coadjoin orbi of A ps p S. If A is a T (F )-formal ype, hen M A = O A. The GL 2 (C)-acion is he usual coadjoin acion, and he momen map µ A is jus resricion of he funcional α o gl 2 (C). The definiion is more complicaed when A is a C(()) formal ype. In his case, le B GL 2 (C) be he upper riangular subgroup. Then, M A = {(Bg, α) (Ad (g)(α)) i O A )} (B\ GL 2 (C)) gl 2 (o). The group GL 2 (C) acs on M A via h(bg, α) = (Bgh 1, Ad (h)α) wih momen map µ A : (Bg, α) α gl2 (C). We can now describe he srucure of M(A, C). Theorem 3. The moduli space M(A, C) is obained as a symplecic reducion of he produc of local daa: ( ) M(A, C) = M Ai 0 GL 2 (C). i Remark 4. For oher varians and a realizaion of he isomonodromy equaions as an inegrable sysem, see [2, 3]. Here, GL 2 (C) acs diagonally on he produc manifold, so ha he momen map µ for he produc is he sum of he momen maps of he facors. Since each facor involves a funcional on gl 2 (o) or gl 2 (C), he definiion of he local momen maps shows ha µ 1 (0) is he se of uples for which he resricions of hese funcionals o gl 2 (C) sum o 0. Wriing each funcional as a 1-form, his is jus he condiion ha he sum of he residues vanish. We conclude his paper wih wo illusraions of he heorem, each wih one irregular singular and one regular singular poin, say a 0 and. Take A s 1 d = diag(a, b) A(T (F ), 1) (so a b) and A e 1 d = ω A(C((ω)), 1). Also, le C be an arbirary nonresonan adjoin orbi. Below, we use he idenificaions gl 2 (o) = gl 2 (C)[ 1 ] d and i = [ω 1 ] d. Under hese idenificaions, he resricion map gl 2 (o) i d has fiber Ce 12 Example 5 (M(A s 0, C )). We firs observe ha Ad (1 + gl 2 (o))(a s ) = A s + ( v 0 u 0 ) d wih u, v C arbirary. Indeed, if X, Y gl 2(C), hen (1+X)Y (1+ X) 1 Y +[X, Y ] (mod 2 ), and he claim follows since ad(diag(a, b))(gl 2 (C)) is he off-diagonal marices. Since GL 2 (o) = GL 2 (C) (1 + gl 2 (o)), we ge {( ) } Ad (GL 2 (o))(a s ) = Ad a 1 u d (GL 2 (C)) v b 1 u, v C. (4) The moduli space is he space of GL 2 (C)-orbis of pairs (α, Y ) wih Y C, and Res(α) + Y = 0. One ( sees from (4) ha every orbi has a represenaive wih α of he form a 1 u for some u, v C. Since T j v b 1 ) is he sabilizer of he leading erm, i follows ha he moduli space is he same as he se of T -orbis of pairs (α, Y ) wih α in his sandard form. We C j d.

6 6 Daniel S. Sage claim ha 2, if C is regular nilpoen M(A s 0, C ) = 1, if C = 0 or C is regular semisimple wih race 0 0, oherwise. To see his, noe ha here are unique represenaives for he T -orbis of sandard α s by aking (u, 1) wih u C, (1, 0), and (0, 0). Each (u, 1) wih u 0 gives rise o Y regular semisimple wih race 0 and deerminan u. The pairs (1, 0) and (0, 1) boh lead o regular nilpoen Y s while (0, 0) jus gives Y = 0. Example 6 (M(A e 0, C )). Here, he moduli space is he space of GL 2 (C)- orbis of riples (Bg, α, Y ), where (Bg, α) M A e, Y C, and Res(α)+Y = 0. This is he same as he space of B-orbis of riples (B, α, Y ). Using I = T I 1, an argumen similar o he one in he previous example shows ha {( ) } Ad (I)(A e ) = Ad z 1 d (T ) 1 z z C. (6) I follows easily ha ( ) z v α = 1 + w d v 1 (7) z for some z, v, w C wih v 0. In fac, each B-orbi has a unique represenaive wih v = 1 and z = 0. This means ha he only adjoin orbis C ha give nonempy moduli space are he orbis of ( ) 0 w 1 0. Thus, M(A e 0, C ) is a singleon if C is regular nilpoen or a regular semisimple wih race zero; oherwise, i is empy. We remark ha in he regular nilpoen case, he unique such connecion is he GL 2 version of he Frenkel-Gross rigid connecion, and his argumen shows ha his connecion is indeed uniquely deermined by is local behavior. Remark 5. By seing C = 0 in hese examples, we obain he corresponding one singulariy moduli spaces: M(A s 0) = 1 and M(A e 0) =. Acknowledgmens I would like o hank Chris Bremer for many helpful discussions and Alexander Schmi for he inviaion o speak a ISAAC (5) References [1] P. Boalch, Symplecic manifolds and isomonodromic deformaions, Adv. Mah. 163 (2001), [2] C. Bremer and D. S. Sage, Moduli spaces of irregular singular connecions, In. Mah. Res. No. IMRN 2013 (2013), [3] C. Bremer and D. S. Sage, Isomonodromic deformaion of connecions wih singulariies of parahoric formal ype, Comm. Mah. Phys., 313 (2012),

7 Regular sraa and moduli spaces of connecions 7 [4] C. Bremer and D. S. Sage, A heory of minimal K-ypes for fla G-bundles, submied o In. Mah. Res. No. IMRN, arxiv: [mah.ag]. [5] C. Bremer and D. S. Sage, Fla G-bundles and regular sraa for reducive groups, arxiv: [mah.ag]. [6] P. Deligne, Équaions différenielles à poins singuliers réguliers, Lecure Noes in Mahemaics, Vol. 163, Springer-Verlag, New York, [7] E. Frenkel and B. Gross, A rigid irregular connecion on he projecive line, Ann. of Mah. (2) 170 (2009), [8] M. Kamgarpour and D. S. Sage, A geomeric analogue of a conjecure of Gross and Reeder, arxiv: [mah.rt]. [9] W. Wasow, Asympoic expansions for ordinary differenial equaions, Wiley Inerscience, New York, Daniel S. Sage Deparmen of Mahemaics Louisiana Sae Universiy Baon Rouge, LA sage@mah.lsu.edu