Jordan Decomposition for Formal G-Connections
aa r X i v : . [ m a t h . C A ] F e b JORDAN DECOMPOSITION FOR FORMAL G -CONNECTIONS MASOUD KAMGARPOUR AND SAMUEL WEATHERHOG
Abstract.
A theorem of Hukuhara, Levelt, and Turrittin states that every formal differential operator hasa Jordan decomposition. This theorem was generalised by Babbit and Varadarajan to the case of formal G -connections where G is a semisimple group. In this paper, we provide straightforward proofs of thesefacts, highlighting the analogy between the linear and differential settings. Contents
1. Introduction 12. Factorisation of differential polynomials 33. Formal differential operators 84. Formal G -connections 12References 131. Introduction
Let K := C (( t )) be the field of formal Laurent series and consider the derivation d : K → K defined by d := t ddt . Let V be a finite dimensional vector space over K . A formal differential operator is a C -linear map D : V → V satisfying the Leibniz rule(1) D ( av ) = aD ( v ) + d ( a ) v, a ∈ K , v ∈ V. It is well-known that linear operators encode linear equations. Similarly, differential operators encode(ordinary) differential equations. Thus, the study of formal differential operators is indispensable in thetheory of meromorphic differential equations; see [Var96] for an extensive review.In analogy with linear operators, differential operators have matrix presentations and it will be convenientto have these at our disposal. Indeed, choosing a basis for V , we can represent D as an operator d + A where A is an n × n matrix with values in K . Changing the basis by an element g ∈ GL n ( K ) amounts to changingthe operator d + A to d + g − Ag + g − dg . Here dg denotes the matrix obtained by applying the derivation d to each entry of the matrix g . The map A g − Ag + g − dg is called gauge transformation and plays an important role in the theory.1.1. Semisimple Connections.
To formulate a Jordan decomposition, we need a notion of semisimplicity.We start with a definition for formal differential operators.
Definition 1.
Let D : V → V be a formal differential operator. Then D is(i) simple if V has no D -invariant subspace(ii) semisimple if every D -invariant subspace has a D -invariant complement(iii) diagonalisable if it has a presentation of the form d + A where A is a diagonal matrix(iv) potentially diagonalisable if it is diagonalisable after a finite base change. Mathematics Subject Classification.
Key words and phrases.
Meromorphic ordinary differential equations, formal differential operators, Hukuhara-Levelt-Turrittin Theorem, Differential polynomials, Differential Hensel’s Lemma, Newton polygons, formal G -connections, canonicalform, Babbit-Varadarajan Theorem. t is easy to show that an operator is semisimple if and only if it is a direct sum of simple ones. Thefollowing theorem gives an explicit description of semisimple operators. Theorem 2 (Levelt) . A formal differential operator is semisimple if and only if it is potentially diagonal-isable.
For future use, we will need the following functorial property. Let D : V → V be a differential operatorand write D = d + A with A ∈ gl ( V ). Consider the adjoint mapad : gl ( V ) → gl ( gl ( V ))Then ad( A ) is a linear operator on gl ( V ); therefore, d + ad( A ) is a differential operator on gl ( V ). Thefollowing observation will be useful. Proposition 3.
The differential operator d + A is semisimple if and only if the differential operator d +ad( A ) : gl ( V ) → gl ( V ) is semisimple. Jordan Decomposition.
We are now ready to discuss the notion of Jordan decomposition.
Theorem 4 (Hukuhara-Levelt-Turrittin) . Every formal differential operator D can be written as a sum D = S + N of a semisimple differential operator S together with a nilpotent K -linear operator N such that S and N commute (as C -linear maps). Moreover, the pair ( S, N ) is unique. The above theorem has numerous applications in the theory of differential operators and other areasof mathematics, cf. [Kat70, Kat87, Luu15, BY15, KS16]. The existence result was first proved by Turrittin[Tur55], building on earlier work of Hukuhara [Huk41]. Turrittin’s argument was rather complicated involvingnine different cases. Subsequently, Levelt gave a more conceptual (albeit still not straightforward) proof andformulated the correct uniqueness statement [Lev75]. As a corollary, he concluded:
Corollary 5.
Every formal differential operator has, after an appropriate finite base change, an eigenvalue.
Levelt asked for a direct proof of this corollary, noting that this would considerably simplify the proofsof the above theorems. Subsequently, several authors provided alternative approaches to these theoremscf. [Was65, Mal79, Rob80, BV83, Pra83, vdPS03, Ked10]. One of our main goals is to provide an elementaryproof of the fact that every differential operator has an eigenvalue and use it to provide a simple proof ofthe existence of Jordan decomposition, thus fulfilling Levelt’s vision.We now provide a brief summary of our approach. Let K { x } denote the non-commutative ring of differ-ential polynomials . As an abelian group K { x } = K [ x ] but multiplication is modified by the rule xa = ax + da for all a ∈ K . Using a version of Hensel’s lemma and Newton polygons, we prove: Theorem 6.
Every non-constant differential polynomial in K { x } has a linear factorisation over a finiteextension of K . The above result is established in §
2. Note that Malgrange [Mal79] and Robba [Rob80] also use Newtonpolygons and differential Hensel’s lemma in their treatment of the Hukuhara-Levelt-Turrittin Theorem;however, our formulation and proof of Jordan decomposition is different from theirs; for instance, we do notuse the cyclic vector lemma. In §
3, we show that Theorem 2 and Corollary 5 follow easily from Theorem 6, thus illustrating the analogiesbetween linear and differential setting. Using these results, we obtain a generalised eigenspace decompositionfor differential operators. In other words, we obtain that every differential operator has a representation d + X where X is a block-upper triangular matrix and each block has a unique (up to similarity) eigenvalue. At thispoint, we encounter a subtle difference between the linear and differential setting. Let us write X = Y + Z where Y is diagonal and Z is strictly upper triangular. If we were considering linear operators, then Y would be the semisimple and Z the nilpotent part of X and these two commute. In the differential setting,however, the situation is more subtle because the operators d + Y and Z do not necessarily commute. Infact, these two operators commute if and only if the entries of Z are complex numbers (i.e. have no powersof t ). We prove that indeed we can arrange so that the entries of Z are complex numbers by using Katz’sclassification of unipotent differential operators [Kat87]. For the advantages and disadvantages of the cyclic vector lemma, cf. [Ked10, § .3. Formal G -connections. The above considerations have a natural generalisation to the setting ofalgebraic groups. Let G be a connected, semisimple, linear algebraic group over C and let g denote its Liealgebra. A formal G -connection is an expression of the form ∇ = d + A, A ∈ g ( K ) := g ⊗ K . The group G ( K ) acts on the space of connections by gauge transformation g · ( d + A ) = d + Ad g ( A ) + ( dg ) g − , g ∈ G ( K ) , A ∈ g ( K ) . One way to make sense of the expression ( dg ) g − is to choose a faithful representation ρ : G → GL n (e.g.the adjoint representation) and show that d ( ρ ( g )) .ρ ( g ) − , a priori in gl n ( K ), actually belongs to g ( K ), andis independent of the chosen representation; see [BV83, § § § Semisimple G -connections. To discuss Jordan decomposition, we first need a notion of semisimplicityfor formal G -connections. Proposition 3 allows us to define such a notion: Definition 7. A G -connection ∇ = d + A , A ∈ g ( K ) , is called semisimple if d + ad( A ) is semisimple (as aformal GL( g ) -connection). The above is analogous to the definition of ad-semisimplicity for elements in a semisimple Lie algebra, cf.[Hum78, § H ⊆ G be a maximal (complex) torus and h := Lie( H ) the corresponding Cartansubalgebra. We then have an analogue of Theorem 2: Theorem 8. A G -connection ∇ = d + A is semisimple if and only if, after a finite base change K ′ / K , ∇ isgauge equivalent to a connection of the form d + X where X ∈ h ( K ′ ) . As far as we know this is the first time the above natural theorem has been formulated in the literature.We use properties of the differential Galois group to establish the above theorem; see § Jordan decomposition.
We are ready to state Jordan decomposition for formal G -connections. Theorem 9 (Jordan decomposition) . Every G -connection ∇ = d + A can be written as a sum ∇ = S + N ,where S is a semisimple G -connection, N ∈ g ( K ) is a nilpotent element and S and N commute. Moreover,the pair ( S, N ) is unique. When we say S and N commute, we mean they commute as elements of the extended loop algebraˆ g = g ( K ) ⊕ C d , where the bracket is defined by[( x ⊗ p ( t ) , α.d ) , ( y ⊗ q ( t ) , β.d )] := [ x ⊗ p ( t ) , y ⊗ q ( t )] + αy ⊗ d ( q ( t )) − βx ⊗ d ( p ( t )) , with x, y ∈ g , p ( t ) , q ( t ) ∈ K , α, β ∈ C .Following a suggestion of Deligne, Babbit and Varadarajan proved an equivalent form of the above the-orem in [BV83]. Their proof, which uses intrinsic properties of algebraic groups, is the only proof of thisfundamental result available in the literature. In this note, we give an alternative proof which uses the ad-joint representation and reduces the problem to the GL n -case. Our approach is thus similar to the standardproofs of (usual) Jordan decomposition for semisimple Lie algebras, cf. [Hum78]. We refer the reader to § Acknowledgment.
We thank Philip Boalch, Peter McNamara, Daniel Sage, Ole Warnaar, and SineadWilson for helpful conversations. We are grateful to Claude Sabbah for sending copies of [Mal79] and [Sab].The material in this paper forms a part of SW’s Master’s Thesis. MK was supported by an ARC DECRAFellowship. 2.
Factorisation of differential polynomials
The goal of this section is to prove Theorem 6. This theorem should be thought of as a differentialanalogue of a classical theorem of Puiseux. As in Section 1, we consider the differential field K := C (( t )) withderivation d . An important implication of Puiseux’s theorem is that for every positive integer b , K b := C (( t b ))is the unique extension of K of degree b . The derivation d extends canonically to a derivation d b on K b .Let R be a C -algebra and d : R → R a derivation. We denote by R { x, d } the ring of differential polynomialsover ( R, d ). We will generally be interested in the cases R = O := C [[ t ]] and R = K := C (( t )) with derivation f the form δ m := t m ddt , for some positive integer m . According to [Ore33], the ring K { x, δ m } is a left andright principal ideal domain.2.1. Differential Hensel’s Lemma.
Let f ∈ O { x, δ m } be a differential polynomial. We write f (mod t n )for the polynomial obtained by first moving all factors of t to the left and then reducing the coefficientsmodulo t n . We denote f (mod t ) by ¯ f . Note that this is a polynomial in C [ x ]. Without loss of generality,we assume throughout that ¯ f = 0.Now suppose we have a factorisation of the form¯ f = g h , g , h ∈ C [ x ] . Our aim is to lift this to a factorisation of f in K { x, δ m } . We think of the following result as a differentialanalogue of Hensel’s lemma. Proposition 10.
Let f ∈ O { x, δ m } and ¯ f = g h as above. Suppose that ( gcd (cid:0) g ( x + n ) , h ( x ) (cid:1) = 1 , ∀ n ∈ Z > if m = 1gcd (cid:0) g ( x ) , h ( x ) (cid:1) = 1 if m > . Then we have a factorisation f = gh with g, h ∈ O { x, δ m } , deg( g ) = deg( g ) , ¯ g = g and ¯ h = h . We note that a version of this proposition appeared in [Pra83, Lemma 1].
Proof.
First of all, in the differential polynomial ring K { x, δ m } , easy induction arguments show that(2) h ( x ) t i = t i h ( x + it m − ) , ∀ h ( x ) ∈ K { x, δ m } , ∀ i ∈ Z , and(3) ( t d x ) k = k − X j =0 a j t kd +( m − j x k − j , ∀ d ∈ Z − { } , ∀ k ∈ N , for some constants a j ∈ C , a = 1.Our goal is to inductively build a sequence of polynomials g n ( x ) = g + tp + t p + · · · + t n − p n − + t n p n , p i ∈ C [ x ](4) h n ( x ) = h + tq + t q + · · · + t n − q n − + t n q n , q i ∈ C [ x ] , (5)which satisfy: f ≡ g n ( x ) h n ( x ) (mod t n +1 ) . If we can do this, then by letting n → ∞ we will obtain elements g, h ∈ O { x, δ m } such that f = gh .Suppose that we know the p i and q i for 1 ≤ i ≤ n −
1. In view of (4) and (5) we have: g n = g n − + t n p n , h n = h n − + t n q n . Requiring that f ≡ g n ( x ) h n ( x ) (mod t n +1 ) then gives us the following condition: f ≡ g n ( x ) h n ( x ) (mod t n +1 ) ≡ (cid:0) g n − ( x ) + t n p n ( x ) (cid:1)(cid:0) h n − ( x ) + t n q n ( x ) (cid:1) (mod t n +1 ) ≡ g n − ( x ) h n − ( x ) + g n − ( x ) t n q n ( x ) + t n p n ( x ) h n − ( x ) + t n p n ( x ) t n q n ( x ) (mod t n +1 ) . We need to shift the powers of t to the left. By (2), g n − ( x ) t n = t n g n − (cid:0) x + nt m − (cid:1) , so we have: f − g n − ( x ) h n − ( x ) ≡ t n g n − (cid:0) x + nt m − (cid:1) q n ( x ) + t n p n ( x ) h n − ( x ) (mod t n +1 ) ≡ t n (cid:0) g n − ( x + nt m − ) q n ( x ) + p n ( x ) h n − ( x ) (cid:1) (mod t n +1 ) , and thus f − g n − ( x ) h n − ( x ) t n ≡ g n − ( x + nt m − ) q n ( x ) + p n ( x ) h n − ( x ) (mod t ) ≡ g ( x + nt m − ) q n ( x ) + p n ( x ) h ( x ) (mod t ) . or notational convenience, we set: f n = f − g n − ( x ) h n − ( x ) t n . so that we have(6) f n ≡ g ( x + nt m − ) q n ( x ) + p n ( x ) h ( x ) (mod t ) . Now if m >
1, then (6) reduces to f n ≡ g ( x ) q n ( x ) + p n ( x ) h ( x ) (mod t ) . Since C [ x ] is a Euclidean domain, we will be able to solve this for p n and q n provided that g and h arecoprime. On the other hand, if m = 1, then (6) becomes f n ≡ g ( x + n ) q n ( x ) + p n ( x ) h ( x ) (mod t ) . In this case, we will only be able to generate the entire sequence if g ( x + n ) and h ( x ) are coprime for all n ∈ Z > .All that remains to show is that we can control the degree of the g n ’s. We will show this in the case m = 1. The proof in the case m > g ( x + n ) with g ( x ) everywhere). Since g ( x + n )and h ( x ) are coprime, we can find a, b ∈ C [ x ] such that g ( x + n ) a ( x ) + h ( x ) b ( x ) = 1 . Multiplying through by f n yields(7) g ( x + n ) a ( x ) f n ( x ) + h ( x ) b ( x ) f n ( x ) = f n ( x ) . Using the division algorithm we can find unique p n and q n such that deg( p n ) < deg( g ). Write: b ( x ) f n ( x ) = Q ( x ) g ( x ) + R ( x )with deg( R ) < deg( g ). Equation (7) then becomes: g ( x + n ) (cid:0) a ( x ) f n ( x ) + Q ( x ) h ( x ) (cid:1) + h ( x ) R ( x ) ≡ f n ( x ) (mod t ) . Setting p n = R and q n = af n + Qh gives us the required g n and h n . (cid:3) Corollary 11.
Let f ∈ O { x, δ } be a monic differential polynomial. Then f admits a factorisation of theform ( x − Λ) h, with Λ ∈ O and h ∈ O { x, δ } .Proof. Let ¯ f ∈ C [ x ] be the reduction of f mod t . Since f is monic, ¯ f is non-constant and hence factors over C into linear factors: ¯ f = ( x − λ )( x − λ ) · · · ( x − λ n ) , λ i ∈ C . Without loss of generality, we can order these factors so that Re( λ ) ≤ Re( λ ) ≤ · · · ≤ Re( λ n ). With thisordering we then have ¯ f = g h , where g = x − λ , h = ( x − λ ) · · · ( x − λ n ) . By our choice of ordering, g ( x + n ) has no common factor with h for all n ∈ Z > . Hence we can applyProposition 10 to obtain a factorisation of the form f = ( x − Λ) h, Λ ∈ O , h ∈ O { x, δ } , as required. (cid:3) Remark . Note that the above result is false for the usual polynomial ring O [ x ]. Indeed, x + t − t does not have a linear factorisation over this ring, but if we consider it as an element of O { x, δ } , then x + t − t = ( x − t )( x + t ). .2. From power series to Laurent series via Newton polygons.
In the previous section, we settledlinear factorisation for differential polynomials in O { x, δ } . In this section, we explain how, by a change ofvariable, we can transform polynomials with coefficients in K to those with power series coefficients. Theprice is that we have to go to a finite extension K q of K and, more seriously, the derivation is not simply thecanonical extension of δ to K q . Nevertheless, we shall see that this change of variable allows us to factorelements of K { x, δ } . Throughout we let v t ( · ) denote the t -adic valuation on K . Lemma 13.
Consider the monic differential polynomial f ( x ) = x n + P ni =1 a i x n − i ∈ K { x, δ } . Let r :=min (cid:8) v t ( a i ) i (cid:9) . Then g ( X ) = t − nr f ( t r X ) is a monic differential polynomial with power series coefficients.Proof. To be more precise, write r = pq with gcd( p, q ) = 1 and q >
0. If r ≥
0, then each v t ( a i ) ≥ f ∈ O { x, δ } . Since we have already dealt with this case in Corollary 11, we may assume that r <
0. Inorder to make the change of variables x = t r X , we require a field extension to K q = C (( t /q )). Let s := t /q so that our change of variables becomes x = s p X . Note that this change of variables means that the relation xt = tx + t becomes Xs q = s q X + s q − p . Hence differential polynomials in X lie in the ring K q { X, q s − p dds } (note this new derivation sends s q to s q − p ).Applying (3) to f ( s p X ) yields f ( s p X ) = s np g ( X ) where g ( X ) = s − np a n + n − X k =0 a k n − − k X j =0 m n − k,j s ( − j − k ) p X n − − k − j , a = 1 . Let v s ( · ) denote the s -adic valuation on K q . Since v s ( a i ) = qv t ( a i ), v t ( a i ) ≥ ipq implies that v s ( a i ) ≥ ip .Thus, for 0 ≤ l ≤ n −
1, the coefficient, b l , of X n − l in g satisfies v s ( b l ) = min ≤ k ≤ l { v s ( a k s − lp ) } ≥ min ≤ k ≤ l { kp − lp } = 0 , where the last equality follows since p < v s ( b l ) will be 0 exactly when v s ( a l ) = lp , that is, if, and only if, v t ( a l ) = lr . For the“constant” term of g we have v s ( b n ) = v s ( a n s − np ) ≥ np − np = 0 , again with equality exactly when v t ( a n ) = nr . Thus g ( X ) = X n + b X n − + · · · + b n , b i ∈ C [[ s ]] , with min( v s ( b i )) = 0. Furthermore, v s ( b i ) = 0 if, and only if, v t ( a i ) = ir . This shows that g ( X ) ∈ C [[ s ]] { X, q s − p dds } . (cid:3) Consider g ( x ) from the above lemma. If ¯ g ( x ) has two distinct roots, then Hensel’s lemma allows us tofactor it. We now study the opposite extreme, i.e., when all roots of ¯ g ( x ) are equal. It will be helpful to usethe notion of Newton polygons for differential polynomials, cf. [Ked10, § r <
0, unless explicitly stated otherwise.
Definition 14 (Newton Polygon) . Let f ∈ K { x, δ m } be a differential polynomial and write f ( x ) = n X i =0 a i x n − i , a i ∈ K . Consider the lower boundary of the convex hull of the points { ( n − i ) , v t ( a i ) : 0 ≤ i ≤ n } ⊂ R . The Newton polygon of f , denoted N P ( f ) , is obtained from this boundary by replacing all line segments ofslope less than − m with a single line segment of slope exactly − m . Lemma 13 now has the following corollary.
Corollary 15.
Let f and g be as in Lemma 13 and suppose that ¯ g := g (mod s ) = ( X + λ ) n , λ ∈ C . Then λ is non-zero and the Newton polygon of f has a single integral slope. roof. As in Lemma 13, write g ( X ) = X n + b X n − + · · · + b n , b i ∈ C [[ s ]] . Since min { v s ( b i ) } = 0, λ = 0. Now since, λ = 0, expanding the bracket ( X + λ ) n shows that v s ( b i ) = 0 forall i and hence v t ( a i ) = ir . Thus, the Newton polygon of f has a single slope of − r and since v t ( a ) = r , r is an integer. (cid:3) For future use, we also record the following lemma.
Lemma 16.
Let f and g be as in Lemma 13 and suppose that ¯ g = ( X + λ ) n , λ ∈ C . Then the slopes of theNewton polygon of f ( x − λt r ) are all strictly smaller than the slope of the Newton polygon of f ( x ) .Proof. By Corollary 15, r is an integer and hence no extension of K is necessary. Since ¯ g = ( X + λ ) n , wecan write g as g = ( X + λ ) n + e ( X + λ ) n − + · · · + e n , e i ∈ O , with v t ( e i ) > i . Now f ( t r X ) = t nr (cid:0) ( X + λ ) n + e ( X + λ ) n − + · · · + e n (cid:1) = ⇒ f ( x ) = t nr (cid:0) ( t − r x + λ ) n + e ( t − r x + λ ) n − + · · · + e n (cid:1) , and hence f ( x − λt r ) = t nr (cid:0) ( t − r x ) n + e ( t − r x ) n − + · · · + e n (cid:1) . Applying (3), we have, for m k,l ∈ C , f ( x − λt r ) = t nr (cid:16) t − nr n − X j =0 m n,j x n − j + e t − ( n − r n − X j =0 m n − ,j x n − − j + · · · + e n (cid:17) = n − X j =0 m n,j x n − j + e t r n − X j =0 m n − ,j x n − − j + · · · + t nr e n . Since v ( e i ) >
0, the valuation of the coefficient of x n − j in f ( x − λt r ) is strictly greater than the correspondingcoefficient in f ( x ). This means that the slopes of the Newton polygon for f ( x − λt r ) are strictly less thanthe slope of the Newton polygon for f ( x ). (cid:3) Example . In order to illustrate Corollary 15 and Lemma 16, consider the differential polynomial f ( x ) = x + (4 t − + 2 t − + 2) x + (4 t − + 4 t − + t − + t − + 1) . In this case, r = − x = t − X yields g ( X ) = X + (4 + 2 t ) X + (4 + 4 t + t + t + t ) , and so ¯ g ( X ) = ( X + 2) . The figure below shows that the Newton polygon of f has only a single slope of − x x + 2 t − as in Lemma 16 yields the new polynomial f ( x ) = x + (2 t − + 2) x + ( t − + t − + 1) . his has a single slope r = − x x − t − yields f ( x ) = x + 2 x + 1. This can easilybe factorised and reversing the change of variables yields the full factorisation f ( x ) = ( x + 2 t − + t − + 1) . y x − − − − − −
11 NP( f )NP( f )NP( f )2.3. Proof of Theorem 6.
Write f ( x ) = x n + a x n − + · · · + a n ∈ K { x, δ } and let r := min (cid:8) v t ( a i ) i (cid:9) ∈ Q .If r ≥
0, then the result follows from the differential Hensel’s Lemma (see Corollary 11) so we may assume r <
0. Let us write r = pq , q > , gcd( p, q ) = 1 . Consider the transformation x t r X . Under this transformation the differential field ( K , δ ) changes to (cid:0) K q , q s − p dds (cid:1) where s := t /q . Moreover, we obtain a monic differential polynomial g ( X ) ∈ C [[ s ]] { y, q s − p dds } .Let ¯ g ( X ) denote the reduction of g ( X ) modulo the maximal ideal of C [[ s ]]. If ¯ g ( X ) has two distinct roots,then we can again apply Proposition 10 to reduce the problem to a polynomial of degree strictly less than f . Thus, we are reduced to the case that ¯ g ( X ) has a unique repeated root λ . For inductive purposes, werename f to f . In this case, by Corollary 15, λ = 0 and the Newton polygon of f has a single integral slope. Now we make the transformation x x − λt r . As shown in Lemma 16, under this transformation f is mapped to a polynomial f whose Newton polygon has slopes strictly less than that of f . Note that thistransformation does not change the differential field.Now we start the process with the polynomial f ( x ) := x n + b x n − + · · · + b n ∈ K { x, δ } ; i.e., we let r := min (cid:8) v ( b i ) i (cid:9) . If r ≥ x t r X to obtaina new polynomial g ( X ). If ¯ g ( X ) has distinct roots, then we are done; otherwise, applying Corollary 15again, we conclude that the Newton polygon of f has a single integral slope. Since the slope of f is anonnegative integer strictly less than slope of f , this process must stop in finitely many steps at which pointwe have a factorisation of our polynomial. (cid:3) Formal differential operators
Recall that for each positive integer b , K b denotes the unique finite extension of K of degree b . Given adifferential operator D , one has a canonical differential operator D ⊗ K K b : V ⊗ K K b → V ⊗ K K b called the base change of D to K b . All base changes considered in this article are of this form. Henceforth,we will use the notation V b := V ⊗ K K b and D b = D ⊗ K K b . .1. Proof of Corollary 5 (Every differential operator has an eigenvalue).
The argument proceedsexactly as in the linear setting. Let D : V → V be a differential operator and v ∈ V be a non-zero vector.Consider the sequence v, D ( v ) , D ( v ) , · · · . As V has finite dimension over K , we must have that D n ( v ) + a D n − ( v ) + · · · + a n − D ( v ) + a n v = 0 , a i ∈ K , where n = dim K ( V ). Now consider the polynomial f ( x ) = x n + a x n − + · · · + a n in the twisted polynomialring K { x } . After a finite extension, we can write f ( x ) = ( x − Λ ) · · · ( x − Λ n ) ∈ K b { x } , Λ i ∈ K b , b ∈ Z > . Thus, ( D b − Λ ) · · · ( D b − Λ n ) v = 0 . Let i ∈ { , , · · · , n } be the largest number such that ( D b − Λ i ) · · · ( D b − Λ n ) v = 0. If i = n , then v is aneigenvector of D b with eigenvalue Λ n . Otherwise ( D b − Λ i +1 ) · · · ( D b − Λ n ) v is an eigenvector of D b witheigenvalue Λ i . (cid:3) Proof of Theorem 2 (Semisimple operators are diagonalisable).
We need the following lemma.The proof is an easy argument using the Galois group Gal( K b / K ); see [Lev75, § Lemma 18. D is semisimple if and only if D b is. Now we are ready to prove Theorem 2. Suppose D is semisimple. We prove by induction on dim( V )that, after an appropriate base change, it is diagonalisable. If dim( V ) = 1 the result is obvious, so assumedim( V ) >
1. Without loss of generality, assume D has an eigenvector v (if not, do an appropriate basechange; by the previous lemma, the operator remains semisimple). Let U = span K { v } . Then U is a one-dimensional, D -invariant subspace of V ; thus, there exists a D -invariant complement W . Now D : W → W is semisimple so by our induction hypothesis (after an appropriate base change), we can write W as a directsum of one-dimensional subspaces. Thus, after an appropriate base change, we have a decomposition of ourvector space into one-dimensional, invariant subspaces and so D is diagonalisable.Conversely, suppose D b is a diagonalisable operator. Then clearly D b is semisimple and thus, by Lemma18, so is D . (cid:3) Invariant Properties of Differential Operators.
The goal of this section is to prove Proposition3. To this end, we need to establish some properties of differential operators.3.3.1.
Invariant Subspaces.
Lemma 19.
Let D : V → V be a differential operator with Jordan decomposition D = S + N . Suppose that W ⊂ V is a D -invariant subspace. Then W is also S -invariant.Proof. Note that V decomposes into generalised eigenspaces and that these generalised eigenspaces are D , S and N invariant [Lev75, § V itself is a generalised eigenspace.In this case, there exists a finite extension, K b , of K such that S = d + λI for some λ ∈ K b . We first provethe result in the case of unipotent differential operators (i.e. in the case λ = 0). As in Section 3.5, we denoteby U the category of unipotent differential operators. Recall this category is equivalent to the category Nilpwhose objects are pairs ( V , N ) where V is a C -vector space and N is a nilpotent linear operator.The restriction D | W : W → W gives us a monomorphism in the category U . Under the equivalence F we obtain a monomorphism in Nilp. Hence, there is a basis of V for which we can write D = d + N . Since( W , N ′ ) ֒ → ( V , N ), in this basis we have dW = W . That is, W is S -invariant.This result clearly extends to differential operators with a unique (up to similarity) eigenvalue.For the general case, recall that after a finite extension to K b , we can write D = S + N where S isdiagonalisable. Now W b is a D b -invariant subspace of V b and so by the above, W b is also S b -invariant. If W were not S -invariant, then W b would not be S b -invariant, hence W must be S -invariant. (cid:3) Indeed, if Dv = λv , λ ∈ K , and a ∈ K then D ( av ) = ( aλ + d ( a )) v ∈ U . .3.2. Adjoint differential operator.
Let V be a finite dimensional vector space over K . Given a differentialoperator d + A : V → V , we write d + S + N for its Jordan decomposition. Note that S is not necessarily asemisimple linear operator on V ; rather, d + S is a semisimple differential operator on V . Lemma 20.
Let d + A : V → V be a differential operator, where A ∈ gl ( V ) . Then d + ad S + ad N is theJordan decomposition of d + ad A .Proof. There exists a finite extension K b of K such that we can pick a basis for V ⊗ K b to put d + A inJordan normal form. In this case, S is diagonal and N is a constant nilpotent matrix with 1’s or 0’s on thesuper-diagonal, and S and N commute. Thus, d + ad( S ) is a semisimple differential operator on gl ( V ⊗ K b ).We claim that it commutes with ad( N ). Indeed,[ d + ad( S ) , ad( N )] = [ d, ad( N )] + [ad( S ) , ad( N )] , where the bracket is for the extended Lie algebra [ gl ( g ). Now ad( N ) is constant, so the first bracket is zero.Since S and N commute, the second bracket is also zero. (cid:3) Proof of Proposition 3. If d + A is semisimple, then we have seen that so is d + ad( A ). If d + A is notsemisimple, then suppose d + S + N is its Jordan decomposition. By assumption, N = 0. This implies thatad N is not trivial. Thus, d + ad A is not semisimple.3.4. Generalised eigenspace decomposition.
Let D : V → V be a formal differential operator and let a ∈ K . Definition 21.
The generalised eigenspace V ( a ) of D is defined as V ( a ) := span K { v ∈ V | ( D − a ) n v = 0 , for some positive integer n . } The goal of this section is to prove the following theorem.
Theorem 22 (Generalised eigenspace decomposition) . For some finite extension K b of K there exists acanonical decomposition V b = L i V b ( a i ) . Moreover, V b ( a i ) ∩ V b ( a j ) = { } ⇐⇒ a i is similar to a j ⇐⇒ V b ( a i ) = V b ( a j ) . Before proving this theorem, we need to recall some facts about differential operators. Let D : V → V bea differential operator. Define H ( V ) := ker( D ) ,H ( V ) := V /D ( V ) . Note that these are vector spaces over C (not over K ). The following proposition due to Malgrange [Mal74,Theorem 3.3] is an analogue of the rank-nullity theorem for formal differential operators. Proposition 23.
Let D : V → V be a formal differential operator. Then dim C H ( V ) = dim C H ( V ) . Next, recall that the dual differential operator D : V → V is the operator D ∗ on the vector space V ∗ = Hom K ( V, K ) defined by D ∗ : V ∗ → V ∗ , D ∗ ( f ) = d ◦ f − f ◦ D, f ∈ V ∗ . Let D : V → V and D ′ : V ′ → V ′ be differential operators. Then, we can define a differential operator D ⊗ D ′ on V ⊗ V ′ by ( D ⊗ D ′ )( v ⊗ v ′ ) := D ( v ) ⊗ v ′ + v ⊗ D ′ ( v ′ ) . The set of all of K { x } -linear maps from V to V ′ is denoted Hom K { x } ( V, V ′ ). This is a C -vector space.The Yoneda extension group Ext K { x } ( V, V ′ ) consists of equivalence classes of extensions of K { x } -modules0 → V → V ′′ → V ′ → K { x } -linear isomorphism between them inducingthe identity on V and V ′ . roposition 24. Let D : V → V and D ′ : V ′ → V ′ be two formal differential operators. Then, we have(i) dim C Ext K { x } ( V, V ′ ) = dim C H ( V ∗ ⊗ V ′ ) .(ii) If no eigenvalue of D is similar to an eigenvalue of D ′ , then Ext K { x } ( V, V ′ ) = 0 .Proof. One can show (see [Ked10, Lemma 5.3.3]) that there is a canonical isomorphism of C -vector spaces:Ext K { x } ( V, V ′ ) ≃ H ( V ∗ ⊗ V ′ ) . This fact together with Proposition 23 implies (i).The eigenvalues of D ∗ ⊗ D ′ are of the form − a + a ′ where a and a ′ are eigenvalues of D and D ′ , respectively.By assumption, − a + a ′ is never similar to zero; thus, kernel of D ∗ ⊗ D ′ is trivial. Part (ii) now follows fromPart (i). (cid:3) Proof of Theorem 22.
We may assume, without the loss of generality, that all eigenvalues of D are already in K (if not, do an appropriate base change). We use induction on dim( V ) to prove the theorem. If dim( V ) = 1then the claim is trivial. Suppose dim( V ) >
1. Then by assumption D has an eigenvector. Hence, we havea one-dimensional invariant subspace U ⊂ V . Let W := V /U . Then D defines a differential operator on W .Moreover, V ∈ Ext K { x } ( U, W ). By induction we may assume that W decomposes as W = M i W ( a i ) , a i ∈ K , for non-similar a i . Now V ∈ Ext K { x } (cid:16) U, M i W ( a i ) (cid:17) ≃ M i Ext K { x } ( U, W ( a i )) . If the eigenvalue a of D | U is not similar to any a i then by the above proposition all the extension groupsare zero, and so V = W ⊕ U and the theorem is established. If a is similar to a j , for some j , then theonly non-trivial component in the above direct sum is Ext K { x } ( U, W ( a j )). But it is easy to see that alldifferential operators in Ext K { x } ( U, W ( a j )) have only a single eigenvalue a j (up to similarity). Hence V hasthe required decomposition. (cid:3) Unipotent differential operators.
Theorem 22 implies that we only need to prove Jordan decom-position for differential operators with a unique eigenvalue. By translating if necessary, we can assume thiseigenvalue is zero. Thus, we arrive at the following:
Definition 25 (Unipotent Operators) . A differential operator is unipotent if all of its eigenvalues are similarto zero.
We now give a complete description of unipotent differential operators. Let Nilp C denote the categorywhose objects are pairs ( V, N ) where V is a C -vector space and N is a nilpotent endomorphism. Themorphisms of Nilp C are linear maps which commute with N . Let U be the category of pairs ( V, D ) consistingof a vector space
V / K and a unipotent differential operator D : V → V . Define a functor F : Nilp C → U , ( V, N ) ( K ⊗ C V, d + N ) . The following result appears (without proof) in [Kat87, § Lemma 26.
The functor F defines an equivalence of categories with inverse given by G : U → Nilp C , ( V, D ) (cid:0) ker( D dim K ( V ) ) , D (cid:1) . Proof.
We first show that the composition G ◦ F equals the identity. Let ( V, N ) ∈ Nilp C with n := dim( V )and consider F ( V, N ) = ( V ⊗ K , d + N ). The kernel of the operator ( d + N ) n acting on V ⊗ K is the set ofall constant vectors. This is an n -dimensional C -vector space. Since d acts as 0 on this space, applying G to( K ⊗ V, d + N ) recovers the pair ( V, N ).Next, let D : V → V be a unipotent differential operator and let n := dim K ( V ). We first show byinduction that ker( D n ) contains n , K -linearly independent vectors. If n = 1 this is obvious. If n >
1, thenthere exists v ∈ V such that Dv = 0. Set U := span K { v } and consider the differential module V /U . Thishas dimension n − { v , . . . , v n } K -linearly independent vectors in ker( D n − ).For each v i we have D n − v i + U = U and hence D n − v i = a i v for some a i ∈ K . Now observe that we can hoose b i such that d n − ( b i ) = a i − a i, where a i, is the constant term of a i ; since we can always “integrate”elements with no constant term. Now we have D n − ( v i − b i v ) = D n − v i − D n − ( b i v ) = a i v − n − X j =0 (cid:18) n − j (cid:19) d j ( b i ) D n − − j ( v ) = a i v − d n − ( b i ) v = a i, v. Hence D n ( v i − b i v ) = D ( a i, v ) = 0 so { v, v − b v, . . . , v n − − b n − v } is a set of K -linearly independentvectors in ker( D n ).Note the functor G sends V to the C -vector space W := ker( D n ) = span C { v, v − b v, . . . , v n − − b n − v } .Moreover, D induces a C -linear operator N on W . By construction, this operator is nilpotent and for thisbasis, the matrix of N is constant (i.e., its entries belong to C ). Applying the functor F to ( W, N ) nowrecovers the differential module (
V, D ). (cid:3) Remark . A formal differential operator D is said to be regular singular if it has a matrix representationof the form A + A t + · · · , A i ∈ gl n ( C ) . It is known that, in this case, D can actually be represented by a constant matrix; i.e., by a matrix A ∈ gl n ( C ).The conjugacy class of A is uniquely determined by D and is called the monodromy [BV83, § Proof of Theorem 4 (Jordan Decomposition).
The uniqueness part of the theorem is relativelyeasy. Since we don’t have anything new to add to Levelt’s original proof, we refer the reader to [Lev75] forthe details. It remains to prove existence.Let D : V → V be a formal differential operator. By Theorem 22, there exists a positive integer b suchthat D b : V b → V b admits a generalised eigenspace decomposition. Thus, D b can be represented by a blockdiagonal matrix where each block is upper triangular with a unique (up to similarity) eigenvalue. Thus, wemay assume without the loss of generality that D b has a unique, up to similarity, eigenvalue a . Replacing D b by D b − a , we may assume that D b is unipotent in which case the result follows from Lemma 26. Thisproves the existence of Jordan decomposition for D b .We now show that the Jordan decomposition of D b descends to a decomposition of D . The proof is similarto the linear setting. Picking a K -basis of V and extending it to a basis of V b allows us to write D b = d + A where A is a matrix with entries in K . Let S b = d + B and N b = C for matrices B and C with respectto this basis. Then, for any σ ∈ Gal( K b / K ), it is clear that d + A = d + σ ( B ) + σ ( C ) is a second Jordandecomposition of D b . Thus, we must have C = σ ( C ) and σ ( B ) = B . Hence, d + B and C are defined over K . (cid:3) Formal G -connections Description of semisimple G -connections. We start by recalling basic facts about the differentialGalois group. Let I K denote the differential Galois group of K as defined in [Kat87, § G -connection ∇ , we get a homomorphism ρ ∇ : I K → G . The Zariski closure of the image of ρ ∇ isan algebraic subgroup of G called the differential Galois group of ∇ and denoted by G ∇ . For an alternativepoint of view on G ∇ , cf. [vdPS03, § Proof of Theorem 8.
We are now ready to prove the theorem. Suppose the differential operator d + A , A ∈ g ( K ), is gauge equivalent to d + X with X ∈ h ( K ′ ) for some finite extension K ′ of K . Then ad( X ) ∈ ad( h )( K ′ ) and ad( h ) is contained in some Cartan subalgebra of gl ( g ). Then there exists g ∈ GL( g )( C ) suchthat g − ad( h ) g ∈ d where d consists of diagonal matrices in gl ( g ). As dg = 0, the gauge action of g on d + ad( X ) yields d + g − ad( X ) g . Thus d + ad( X ) is gauge equivalent to a diagonal differential operatorand is therefore semisimple by Theorem 2. As d + ad( X ) is gauge equivalent to d + ad( A ), this implies that d + ad( A ) is semisimple. By definition, then d + A is semisimple.Conversely, suppose that d + A is semisimple, i.e. d + ad( A ) is semisimple. By Theorem 2, d + ad( A ) isdiagonalizable after a finite extension K ′ of K . This implies that the image of the composition I K ′ → G → GL( g ) s a subgroup of a torus in GL( g ). This then implies that the image of I K ′ → G is a subgroup of a maximaltorus H ⊂ G ; that is, the above map factors through a map I K ′ → H . Thus, d + A is equivalent to aconnection of the form d + X for some X ∈ h ( K ′ ). (cid:3) Remark . Let ∇ = d + X be a semisimple formal G -connection. By Theorem 8, we may assume (after afinite base change) that X ∈ h ( K ) where h is a Cartan subalgebra of g . Write X = P i X i t i where X i ∈ h ( C )and set X + = X i ≥ X i t i ∈ h [[ t ]] , X − = X i ≤ X i t i ∈ h [ t − ] . Let A := exp (cid:16) − Z X + t dt (cid:17) ∈ H ( C [[ t ]]) . Then gauge transformation of ∇ by A yields d + X − . This is the canonical form of ∇ in the sense of [BV83].4.2. Jordan decomposition for G -connections. We start with a lemma, which is an analogue of astandard result in Lie theory, c.f. [Hum78, § Lemma 29.
Let g ⊂ gl ( V ) be a Lie subalgebra. Let d + A : V → V , A ∈ g ( K ) , be a differential operatorwith Jordan decomposition (as a GL( V ) -connection) D = d + X + N . Then X ∈ g ( K ) ; moreover, d + X isa semisimple G -connection.Proof. By definition g ( K ) is a ( d + ad A )-invariant subspace of gl ( V ) ⊗ K . Thus, by Lemma 19, it is also( d + ad X )-invariant. By definition, g ( K ) is d invariant. Thus, g ( K ) is ad X -invariant. This implies that( d + ad X ) − d : g ( K ) → g ( K ) is a K -linear derivation on g ( K ) and hence X ∈ g ( K ) (since every K -linearderivation is inner). (cid:3) Proof of Theorem 9.
We are now ready to prove the theorem. Let ∇ = d + A be a G -connection. Note thatthe adjoint action gives an embedding g ( K ) ⊂ gl ( g ( K )). Let d + X + N denote the Jordan decompositionof d + A as a differential operator g ( K ) → g ( K ). Then by the previous lemma, X ∈ g ( K ) and d + X is asemisimple G -connection. It follows that N = d + A − ( d + X ) is a nilpotent element of g ( K ). Now d + X and N commute in the extended loop algebra of gl ( g ( K )). This implies that they commute in the extendedloop algebra of g ( K ), this establishes the existence of Jordan decomposition.For uniqueness, suppose d + X + N and d + X + N are Jordan decompositions for ∇ . Then d +ad( X ) + ad( N ) and d + ad( X ) + ad( N ) are Jordan decompositions for ad( ∇ ). By uniqueness of Jordandecomposition for differential operators (Theorem 4), we obtain ad( X ) = ad( X ) and ad( N ) = ad( N ).As the adjoint representation is faithful, we conclude X = X and N = N . (cid:3) Remark . A formal G -connection ∇ is called unipotent if its semisimple part is trivial. One can show that ∇ is unipotent if and only if its differential Galois group G ∇ is unipotent. According to [vdPS03, thm. 11.2], G ∇ is then a one-parameter subgroup of G generated by a unipotent element. This implies that the map Y d + Y from nilpotent elements of g ( C ) to formal unipotent G -connections defines a bijection between nilpotentorbits in g ( C ) and equivalence classes of unipotent connections. Thus, one obtains a generalisation of Katz’sresults (Lemma 26) to the setting of G -connections. References [BV83] B. Babbitt and P. Varadarajan, formal reduction theory of formal differential equations: a group theoretic view ,Pacific J. Math. (1983), no. 1, 1–80.[BY15] P. Boalch and D. Yamakawa,
Twisted wild character varieties , arXiv:1512.08091 (2015).[Fre07] Edward Frenkel,
Langlands Correspondence for Loop Groups , Cambridge Studies in Advanced Mathematics, vol. 103,Cambridge University Press, Cambridge, 2007.[Huk41] M. Hukuhara,
Th´eor`emes fondamentaux de la th´eorie des ´equations diff´erentielles ordinaires. II , Mem. Fac. Sci.Ky¯usy¯u Imp. Univ. A. (1941), 1–25.[Hum78] J. E. Humphreys, Introduction to Lie Algebras and Representation Theory , Graduate Texts in Mathematics, vol. 9,Springer-Verlag, New York-Berlin, 1978. Second printing, revised.[Kac90] V. Kac,
Infinite-dimensional Lie algebras , 3rd ed., Cambridge University Press, Cambridge, 1990.[KS16] M. Kamgarpour and D. Sage,
A geometric analogue of a conjecture of Gross and Reeder , arXiv:1606.00943 (2016). Kat70] Nicholas M. Katz,
Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin , Inst.Hautes ´Etudes Sci. Publ. Math. (1970), 175–232.[Kat87] N. M. Katz, On the calculation of some differential Galois groups , Invent. Math. (1987), no. 1, 13–61.[Ked10] K. S. Kedlaya, p -adic differential equations , Cambridge Studies in Advanced Mathematics, vol. 125, CambridgeUniversity Press, Cambridge, 2010.[Lev75] G. Levelt, Jordan decomposition for a class of singular differential operators , Ark. Mat. (1975), 1–27.[Luu15] M. Luu, Local Langlands duality and a duality of conformal field theories , arXiv:1506.00663 (2015).[Mal74] B. Malgrange,
Sur les points singuliers des ´equations diff´erentielles , Enseignement Math. (2) (1974), 147–176.[Mal79] , Sur la r´eduction formelle des ´equations diff´erentielles `a singularit´es irr´eguli`eres (1979).[Ore33] O. Ore,
Theory of non-commutative polynomials , Ann. of Math. (2) (1933), no. 3, 480–508.[Pra83] C. Praagman, The formal classification of linear difference operators , Indagationes Mathematicae (Proceedings) (1983), no. 2, 249 - 261.[Ras15] S. Raskin, On the notion of spectral decomposition in local geometric Langlands , arXiv:1511.01378 (2015).[Rob80] P. Robba,
Lemmes de Hensel pour les op´erateurs diff´erentiels. Application `a la r´eduction formelle des ´equationsdiff´erentielles , Enseign. Math. (2) (1980), no. 3-4, 279–311 (1981).[Sab] C. Sabbah, Introduction to algebraic theory of linear systems of differential equations , Unpublished lecture notes.[vdPS03] M. van der Put and M. F. Singer,
Galois theory of linear differential equations , Grundlehren der MathematischenWissenschaften [Fundamental Principles of Mathematical Sciences], vol. 328, Springer-Verlag, Berlin, 2003.[Tur55] H. L. Turrittin,
Convergent solutions of ordinary linear homogeneous differential equations in the neighborhood ofan irregular singular point , Acta Math. (1955), 27–66.[Var96] V. S. Varadarjan, Linear formal differential equations: a modern point of view , Bulletin of AMS (1996), no. 1.[Was65] W. Wasow, Asymptotic expansions for ordinary differential equations , Pure and Applied Mathematics, Vol. XIV,Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1965.