From Whittaker Functions to Quantum Groups by Daniel ump Supported in part by NSF grant 1001079 1
etween Physics and Number Theory We will describe a new connection between existing topics in Mathematical Physics and Number Theory. In Physics, Solvable Lattice Models are an important example in Statistical Mechanics. The key to their secrets is the Yang-axter equation. In Number Theory Whittaker functions appear in the Fourier coefficients of automorphic forms. Particularly Metaplectic Whittaker functions are somewhat mysterious. The key to their properties are intertwining integrals which relate different principal series representations of covers of GL(r, Q p ). We will connect these topics, reviewing two papers: Metaplectic Ice (2012) by rubaker, ump, Friedberg, hinta, Gunnells Yang-axter equation for Metaplectic Ice (2016) by uciumas, rubaker, ump 2
Overview If H is a quantum group (quasitriangular Hopf algebra) and if U, V are modules, then there is an isomorphism U V V U making the modules into a braided category. This isomorphism is described by an R-matrix. particular quantum group is H = U q (gl ˆ(n)). For z it has an n-dimensional module V (z). If F is a nonarchimedean local field containing the group µ 2n of n-th roots of unity, there is a metaplectic n-fold cover of GL(r, F). It is a central extension: 0 µ 2n GL(r, F) GL(r, F) 0. The spherical representations are parametrized by a Langlands parameter z = (z 1,, z n ). There is a vector space W z of Whittaker functions and standard intertwining maps W z W w(z) (w W, the Weyl group). We will explain that W z V (z @ 1 ) V (z n ) and the intertwining maps are described by R-matrices. The supersymmetric quantum group U q (gl ˆ(n 1)) also plays a role. 3
Part I: Quantum Groups 4
Statistical Mechanics In Statistical Mechanics, each state s i of a system S with energy E i has probability proportional to e ke i. The exact probability is Z 1 e ke i where Z = e ke i (k=oltzmann s constant) s i S is the partition function. In this setting it is real valued, but: In 1952, Lee and Yang showed that sometimes the partition function may be extended to an analytic function of a complex parameter. Its zeros lie on the unit circle. This is like a Riemann hypothesis. We will represent p-adic Whittaker functions as complex analytic partition functions. The Langlands parameters are in. Solvable Lattice Models In some cases the partition function may be solved exactly. These examples are important for studying phase transitions. Unfortunately all known examples are two-dimensional. 5
Solvable Lattice Models In these examples, the state of a model is described by specifying the values of parameters associated with the vertices or edges of a planar graph. The Ising Model was solved by Onsager in 1944. The six-vertex model was solved by Lieb, Sutherland and axter in the 1960 s. The six-vertex model is a key example in the history of Quantum Groups. key principle underlying these examples is the Yang-axter equation. (This is not one equation but a class of equations.) The algebraic context for understanding Yang-axter equations is quantum groups. Once the theory of quantum groups is in place, many examples of Yang-axter equations present themselves. For us the important ones are associated with the quantum groups U q (gl(n)) and U q (gl(n 1)). 6
Two Dimensional Ice egin with a grid, usually (but not always) rectangular: Each exterior edge is assigned a fixed spin or. The inner edges are also assigned spins but these will vary. state of the system is an assignment of spins to the interior edges The allowed configuration of spins around a vertex has six allowed possibilities 7
Six Vertex Model state of the model is an assignment of spins to the inner edges. (The outer edges have preassigned spins. Every vertex is assigned a set of oltzmann weights. These depend on the spins of the four adjacent edges. For the six-vertex model there are only six nonzero oltzman weights: a 1 a 2 b 1 b 2 c 1 c 2 The oltzmann weight of the state is the product of the weights at the vertices. The partition function is the sum over the states of the system. 8
axter In the field-free case let S be oltzmann weights for some vertex with a = a 1 = a 2 and b = b 1 = b 2 and c = c 1 = c 2. Following Lieb and axter let S = a2 b 2 c 2. 2ab Given one row of ice with oltzmann weights S at each vertex: δ 1 δ 2 δ 2 δ 2 Here δ i, ε i are fixed. µ S S S S ν Toroidal domain: µ is considered an interior edge after gluing the left and ǫ 1 ǫ 2 ǫ 3 ǫ 4 right edges. So we sum over µ and the other interior edges. Let δ = (δ 1, δ 2, ) and ε = (ε 1, ε 2, ) be the states of the top and bottom rows. The partition function then is a row transfer matrix Θ S (δ, ε). The partition function with several rows is a product transfer matrices. Theorem 1. (axter) If S = T then Θ S and Θ T commute. (This is closely related to solvability of the model.) 9
The Yang-axter equation Theorem 2. (axter) Let S and T be vertices with field-free oltzmann weights. If S = T then there exists a third R with R = S = T such that the two systems have the same partition function for any spins δ 1, δ 2, δ 3, ε 1, ε 2, ε 3. ǫ 3 ǫ 3 ǫ 2 S δ 1 ǫ 2 T δ 1 R R ǫ 1 T δ 2 ǫ 1 S δ 2 δ 3 δ 3 Remember: R = a2 b 2 c 2. 2ab 10
ommutativity of Transfer Matrices This is used to prove the commutativity of the transfer matrices as follows. onsider the following system, whose partition function is the product of transfer matrices Θ S Θ T (φ, δ)= ε Θ S(φ, ε)θ T (ε, δ): φ 1 φ 2 φ 3 φ 4 µ S S S S 1 µ 1 µ T T T T 2 µ 2 δ 1 δ 2 δ 3 δ 4 Toroidal oundary onditions: µ 1, µ 2 are interior edges and so we sum over µ 1, µ 2. Insert R and another vertex R 1 that undoes its effect: 11
φ 1 φ 2 φ 3 φ 4 µ 1 R 1 R S S S S µ 1 µ 2 T T T T µ 2 δ 1 δ 2 δ 3 δ 4 Now use YE repeatedly: φ 1 φ 2 φ 3 φ 4 T T T T µ 1 R 1 µ 1 µ 2 S S S S µ2 δ 1 δ 2 δ 3 δ 4 R Due to toroidal now R, R 1 are adjacent again and they cancel. The transfer matrices have been shown to commute. 12
raided ategories monoidal category (Maclane) has associative operation. In symmetric monoidal category (Maclane) we assume: oherence: any two ways of going from one permutation of 1 2 to another give the same result. τ, Important generalization! 1 τ, τ, 1 Maclane assumed that τ, : and τ, : are inverses. ut... Joyal and Street proposed eliminating this assumption. This leads to the important notion of a braided monoidal category. 13
In a symmetric monoidal category τ, : and τ 1, : same. In a braided monoidal category they may be different. are the τ, τ, 1 Represent them as braids. The top row is The bottom row is oherence: ny two ways of going from 1 2 to itself gives the same identity provided the two ways are the same in the rtin braid group. These diagrams describe two morphisms. The morphisms are the same since the braids are the same. 14
Quantum Groups Hopf algebras are convenient substitutes essentially generalizations of the notion of a group. The category of modules or comodules of a Hopf algebra is a monoidal category. The group G has morphisms µ: G G G and : G G G, namely the multiplication and diagonal map. These become the multiplication and the comultiplication in the Hopf algebra. The modules over a group form a symmetric monoidal category. There are two types of Hopf algebras with an analogous property. In a cocommutative Hopf algebra, the modules form a symmetric monoidal category. In a commutative Hopf algebra, the comodules form a symmetric monoidal category. quantum group is a Hopf algebra whose modules (or comodules) form a braided monoidal category. The axioms needed for this define a quasitriangular Hopf algebra (Drinfeld). These are quantum group. 15
The R-matrix a a b b c c Field-free oltzmann weights From this point of view, the oltzmann weights at a field-free vertex R go into a matrix, an endomorphism of V 1 V 2, where V 1 and V 2 are two-dimensional vector spaces, each with a basis labeled v and v. ε 2 ε 1 ε 3 R ε 3 ε 4 ε 1 ε 2 ε 4 We interpret the vertex as a linear transformation R(v ε1 v ε2 ) = ε R 3 ε 4 ε1 ε 2 (v ε3 v ε4 ) If τ: V 1 V 2 V 2 V 1 is τ(x y)= y x Then τr: V 1 V 2 V 2 V 1 is a candidate for a morphism in a braided monoidal category 16
raid Picture Given R,S,T with R = S = T, rotate Yang-axter equation picture: U V W U V W R T S S T R W V U W V U So associating with R, S, T two dimensional vector spaces and interpreting τr: U V V U as a morphism in a suitable category, the Yang- axter equation means the category is braided. Tannakian Program: text Yang axter equation text raided category Quantum Group In the axter case, the relevant quantum group is U q (sl ˆ2). 17
Parametrized Yang-axter equations U V W U V W R T S S T R W V U W V U If we unravel the definitions, the Yang-axter equation means that R 12 S 13 T 23 = T 23 S 13 R 12 where R 12 is the endomorphism R I W acting on the first and third components of U V W and trivially on W, etc. parametrized YE gives an R-matrix R(γ) for every element γ of a group Γ such that R 12 (γ)r 13 (γδ)r 23 (δ) = R 23 (δ)r 13 (γδ)r 12 (γ). The axter YE discussed gives a parametrized equation with Γ =. 18
ffine quantum groups Where do parametrized Yang axter equations come from? Given a Lie group G with Lie algebra g we can try to build a deformation of the universal enveloping algebra U(g). This is U q (g), invented by Drinfeld and Jimbo after the example g = sl 2 was found by Kulish, Reshetikhin and Sklyanin. The quantum group U q (g) will have modules corresponding to the irreducible representations of G. The untwisted affine Lie algebra ĝ was constructed first by physicists, then appeared as a special case of the Kac-Moody Lie algebras. It is a central extension 0 c ĝ g [t, t 1 ] 0. Given a G-module V, U q (ĝ) will have one copy of V for every z. The corresponding Yang-axter equation will be a parametrized one. For the axter example, g = sl 2 and z corresponds to (a, b, c). The parameter q is chosen so = 1 2 (q q1 ). 19
Part II: Tokuyama s formula and the asselman-shalika formula 20
Whittaker models Let F be a nonarchimedean local field, G a reductive algebraic group like GL(r). Langlands associated another algebraic group Ĝ (the connected L- group) such that very roughly (some) representations of G(F) are related to the conjugacy classes of Ĝ(). Let T and Tˆ be maximal tori in G and Ĝ. If z Tˆ() there is a representation π z of G(F) called a spherical principal series. It is infinite-dimensional. If z and z are conjugate then there exists a Weyl group element such that w(z) = z. orrespondingly there is an intertwining operator w : π z π w(z) which is usually an isomorphism. The representation π z may (usually) be realized on a space W z of functions on the group called the Whittaker model. The representation π is infinitedimensional. ut let K be a (special) maximal compact subgroup. Then the space of K-fixed vectors is (usually) one-dimensional. orrespondingly there is a unique K-fixed vector in the Whittaker model. 21
The Weight Lattice Let G and Ĝ be a reductive group and its dual, with dual maximal tori T and Tˆ. Then there is a lattice Λ, the weight lattice which may be identified with either the group X (Tˆ) of characters of Tˆ, or the group T(F)/T(o) where o is the ring of integers in the local field F. There is a cone Λ in Λ consisting of dominant weights. If λ Λ let z z λ (z Tˆ) be λ regarded as a character of Tˆ. Let λ T(F) correspond. Example. If G = GL(r) then Ĝ = GL(r) also. T and Tˆ are the diagonal tori. The weight lattice Λ Z r and if λ = @ (λ 1,,λ r ) then z = z 1 λ1 z λ λ = z 1 λ 1 z r r,, λ=, λr o a prime element. z r 22
The Weyl haracter Formula Let Ĝ() be a reductive Lie group. Irreducible characters χ λ are parametrized by dominant weights λ. The Weyl character formula is (1 z α )χ λ (z) = ( 1) l(w) z w(λρ)ρ ρ = 1 2 α Φ w W The asselman-shalika Formula α Φ α Let F be a nonarchimedean local field and o its ring of integers. Let Ĝ be the Langlands dual of the split reductive group G. Let T and Tˆ be dual maximal tori of G, Ĝ. The weight lattice Λ of Ĝ is in bijection with T(F)/ T(o). Let λ Λ be a dominant weight an t λ be a representative in T(F). The asselman-shalika formula shows that for the spherical Whittaker function W W(t λ ) = { ( ) α Φ (1 q1 z α )χ λ (z) if λ is dominant, 0 otherwise. Here q is the residue field size. The unimportant constant ( ) is a power of q. The expression is a deformation of the Weyl character formula. 23
The Weyl character formula The Weyl character formula can be thought of as a formula for α Φ (1 z α )χ λ (z). λ = dominant weight of some Lie group Ĝ Φ = positive roots χ λ = character of an irreducible representation with highest weight λ Tokuyama s deformation of the WF The Weyl character formula has a deformation (Tokuyama, 1988). This was stated for GL(r) but other artan types may be done similarly. It is a formula for: α Φ (1 vz α )χ λ (z). Tokuyama expressed this as a sum over strict Gelfand-Tsetlin patterns with shape λ ρ. 24
Tokuyama s Formula: rystal Version There are two ways of expressing Tokuyama s formula. α Φ (1 vz α )χ λ (z). s a sum over the Kashiwara crystal based λρ. s a partition function. We will not discuss the first way, except to mention that it was a milestone towards the discovery of the connection between Whittaker functions and quantum groups. rystal bases are combinatorial analogs of representations of Lie groups. In Kashiwara s development, they come from the theory of quantum groups. 25
Tokuyama s Formula: Six Vertex Model There is another way of writing Tokuyama s formula that was first found by Hamel and King. In this view, Tokuyama s formula represents Z(S) = α Φ (1 vz α )χ λ (z) as the partition function of a six-vertex model system Z(S). We may use the following oltzmann weights. We will label this vertex T(z i ) T(z i ) a 1 a 2 b 1 b 2 c 1 1 z i v z i (1 v)z i 1 Here if z 1,, z r are the eigenvalues of z GL(r, ), we take an ice model with r rows, and the above weights at each vertex in the r-th row. (The boundary conditions depend on λ.) These weights are free-fermionic meaning a 1 a 2 b 1 b 2 = c 1 c 2. c 2 26
The R-matrix The Yang-axter equation was introduced into this setting by rubaker, ump and Friedberg. Here is the R-matrix (also free-fermionic): R z1,z 2 z 2 vz 1 z 1 vz 2 v(z 1 z 2 ) z 1 z 2 (1 v)z 1 (1 v)z 2 Yang-axter equation: the following partition functions are equal. It can be used to prove Tokuyama s theorem. ǫ 3 T(z 1 ) T(z 2 ) ǫ 3 ǫ 2 δ 1 ǫ 2 δ 1 R z1,z 2 R z1,z 2 ǫ 1 δ 2 T(z 2 ) ǫ 1 T(z 1 ) δ 2 δ 3 δ 3 27
nother Yang-axter equation We can make a Yang-axter equation involving only the R z1,z 2. The following partition functions are equal. ε 3 ε 4 ε 3 ε 4 R z2,z 3 R z1,z 2 R z1,z 3 ε 2 ε 5 ε 2 ε 5 R z1,z 3 ε 1 R z1,z 2 R z2,z 3 ε 1 ε 6 ε 6 This is a parametrized YE with parameter group. So we expect the corresponding quantum group to be affine. It turns out to be U v (gl ˆ(1 1)). Here gl ˆ(1 1) is a Lie superalgebra. 28
Digression: the Free-Fermionic Quantum Group It was shown by Korepin and by rubaker-ump-friedberg that all free-fermionic oltzmann weights (a 1 a 2 b 1 b 2 = c 1 c 2 ) can be assembled into a parametrized YE with nonabelian parameter group GL(2) GL(1). Thus the weights R z1,z 2 and T(z 1 ), T(z 2 ) all fit into a single YE. This does not generalize to the metaplectic case that we consider next. The corresponding quantum group appears to be new. It has been studied by uciumas. 29
Part II: Metaplectic Ice 30
The Metaplectic Group If F contains the group µ 2n of n-th roots of unity, there is a metaplectic n- fold cover of GL(r, F). It is a central extension: 0 µ 2n GL(r, F) GL(r, F) 0. This fact may be generalized to general reductive groups. Whittaker models are no longer unique. Metaplectic Ice: the Question In 2012, rubaker, ump, hinta, Friedberg and Gunnells showed that Whittaker functions on GL(r, F) could be represented as partition functions of generalized six-vertex models. They were not able to find the R-matrix for reasons that are now finally understood. 31
The partition function a1 a 0 0 a1 a 0 1 0 1 1 0 1 z i g(a) z i (1 v)z i 1 We decorate each horizontal edges spin with an integer modulo n. The spins must be decorated with 0. v = q 1 where q is the cardinality of the residue field. Here g(a) is a Gauss sum. g(a)g( a)=v if a 0 mod n, while g(0) = v. s before, λ is built into the boundary conditions. The partition function equals a metaplectic Whittaker function evaluated at λ. The decorated edges span (Z/2Z)-graded super vector space with n even-graded basis vectors and 1 odd-graded vector. 32
Metaplectic Ice: the nswer In very recent work of rubaker, uciumas and ump: a a a b a a 0 0 a a z 2 n vz 1 n 0 a b a b b g(a b)(z 1 n z 2 n ) (1 v)z 1 c z 2 nc a 0 a a 0 0 z 1 n vz 2 n 0 0 a 0 0 a 0 0 a a v(z n 1 z n 2 ) z n n 1 z 2 (1 v)z a 1 z na 2 (1 v)z na a 1 z 2 c a b. The representatives a and c are chosen between 0 and n. This R-matrix gives the Yang-axter equation for metaplectic ice. 33
Supersymmetric partition function The quantum group is U v (gl ˆ(n 1)) with v = q 1 procedure introduces Gauss sums into the R-matrix. The U v (gl ˆ(n m)) R-matrix was discovered by Perk and Schultz. It was not related to U v (gl ˆ(n m)) until later. It appears in the theory of superconductivity. Intertwining integrals The Whittaker model is no longer one-dimensional. Let W z be the finitedimensional space of spherical Whittaker functions. We recall that the standard intertwining integrals w : W z W w(z) for Weyl group elements w. Theorem 3. (rubaker, uciumas, ump) The quantum group U v (gl ˆ(n)) has one n-dimensional module V (z) for each z. We have an isomorphism W z V (z @ 1 ) V (z n ) that realizes the standard intertwining integrals as U v (gl ˆ(n))-module homomorphisms. Lie superalgebra is not involved in this statement. It is very plausible that an identical statement is true for general artan types. 34