INTEGRATION ON PRODUCT SPACES AND GLn OF A VALUATION FIELD OVER A LOCAL FIELD

We present elements of a theory of translation-invariant integration on finite dimensional vector spaces and on GLn over a valuation field with local field as residue field. We then discuss the case of an arbitrary algebraic group. This extends the work of Fesenko. Introduction This paper addresses the problem of measure and integration on a finite dimensional vector space and on GLn over a valuation field whose residue field is a local field. This, and the more fundamental problem of integration over the field itself have been considered by Fesenko [3] [4] [5] and Kim and Lee [10] [11] in the case of a higher dimensional local field, and by the author [12]. Far more general results of Hrushovski and Kazhdan using model theory [6] [7] treat the case in which F has characteristic zero. Such a theory has applications in the representation theory of two-dimensional local fields (see [10]) and related problems in the Langlands programme. We now outline the contents of the paper. Let F be a valued field with arbitrary value group and ring of integers OF , whose residue field is a non-discrete locally compact field; let C(Γ) be the field of fractions of the complex group algebra of Γ. In [12] the author used ideas of Fesenko [3] [5] to introduce elements of a theory of integration over F with values in C(Γ). In the first section, we give a summary of the results required for this paper. In the second section, the integral on F is extended to F using repeated integration. So that Fubini’s theorem holds, we consider C(Γ)-valued functions f on F such that for any permutation σ of {1, . . . , n} the repeated integral ∫F . . . ∫F f(x1, . . . , xn) dxσ(1) . . . dxσ(n) is well defined, and its value does not depend on σ; such a function is called Fubini. Now suppose that g is a Schwartz-Bruhat function on F n ; let f be the complex-valued function on F which vanishes on O F , and satisfies f(x1, . . . , xn) = g(x1, . . . , xn) for x1, . . . , xn ∈ OF . f is shown to be Fubini in the second section. In proposition 3.11 it is shown that if a ∈ F and τ ∈ GLN (F ), then x 7→ f(a+ τx) is also Fubini and ∫Fn f(a+ τx) dx = | det τ | ∫Fn f(x) dx (∗) 2000 Mathematics Subject Classification 11S80 (primary), 20G25, 20G05, 28C10 (secondary). The author is supported by an EPSRC Doctoral Training Grant at the University of Nottingham. 2 MATTHEW T. MORROW where | · | is an absolute value on F . The main result of the third section, theorem 3.4, easily follows: there exists a space of Fubini functions L(F,GLn) such that L(F,GLn) is closed under affine changes of variable, with (∗) holding for f ∈ L(F,GLn). Next, just as in the classical case of a local field, we look at C(Γ)-valued functions φ on GLn(F ), for which τ 7→ φ(τ)| det τ | belongs to L(F 2 ), having identified F 2 with the space of n×n matrices over F . This leads to an integral on GLn(F ) which is left and right translation invariant, and which lifts the Haar integral on GLn(F ) in a certain sense. Finally we discuss extending the theory to the case of an arbirary algebraic group. Acknowledgements I. Fesenko provided invaluable help during the writing of this text. Notation Let Γ be a totally ordered group and F a field with a valuation ν : F → Γ with residue field F , ring of integers OF and residue map ρ : OF → F (also denoted by an overline). Suppose further that the valuation is split; that is, there exists a homomorphism t : Γ → F such that ν ◦ t = idΓ. Sets of the form a+ t(γ)OF are called translated fractional ideals. C(Γ) denotes the field of fractions of the complex group algebra Γ; the basis element of the group algebra corresponding to γ ∈ Γ shall be written as X rather than as γ. With this notation, XX = X. Note that if Γ is a free abelian group of finite rank n, then C(Γ) is isomorphic to the rational function field C(X1, . . . , Xn). We fix a choice of Haar measure on F . The measure on F × is chosen to satisfy dx = |x|dx, and the measure on F m is always the product measure. Remark. The assumptions above hold for a higher dimensional local field. For basic definitions and properties of such fields, see [8]. Indeed, suppose that F = Fn is a higher dimensional local field of dimension n ≥ 2: we allow the case in which F1 is an archimedean local field. If F1 is non-archimedean, instead of the usual rank n valuation v : F → Z, let ν be the n − 1 components of v corresponding to the fields Fn, . . . , F2; note that v = (νF ◦ η, ν). If F1 is archimedean, then F may be similarly viewed as an valuation field with value group Z and residue field F1. The residue field of F with respect to ν is the local field F = F1. If F is non-archimedean, then the ring of integers OF of F with respect to the rank n valuation is equal to ρ (OF ), while the groups of units O F with respect to the rank n valuation is equal to ρ (O F ). 1. Integration on F In [12] a theory of integration on F taking values in the field C(Γ) is developed. We repeat here the definitions and main results. Definition 1.1. For g a function on F taking values in an abelian group A, set g : F → A x 7→ { g(x) x ∈ OF 0 otherwise. INTEGRATION ON PRODUCT SPACES AND GLn 3 More generally, for a ∈ F , γ ∈ Γ, the lift of g at a, γ is the A-valued function on F defined by g(x) = { g((x− a)t(−γ)) x ∈ a+ t(γ)OF 0 otherwise Note that g = g and g(a+ t(γ)x) = g(x) for all x ∈ F . Definition 1.2. Let L denote the space of complex-valued Haar integrable functions on F . A simple function on F is a C(Γ)-valued function of the form x 7→ g(x)X for some g ∈ L, a ∈ F , γ, δ ∈ Γ. Let L(F ) denote the C(Γ) space of all C(Γ)-valued functions spanned by the simple functions; such functions are said to be integrable on F . Remark 1.3. Note that the space of integrable functions is the smallest C(Γ) space of C(Γ)-valued functions on F with the following properties: (i) If g ∈ L, then g ∈ L(F ). (ii) If f ∈ L(F ) and a ∈ F then L(F ) contains x 7→ f(x+ a). (iii) If f ∈ L(F ) and α ∈ F then L(F ) contains x 7→ f(αx). In fact, it is clear that if f is simple then for a ∈ F and α ∈ F, the functions x 7→ f(x+ a) and x 7→ f(αx) are also simple. The main result on existence and properties of an integral is as follows: Theorem 1.4. There is a unique C(Γ)-linear functional ∫F on L(F ) which satisfies (i) ∫F lifts the usual integral on F : for g ∈ L, ∫F (g) = ∫ g(u) du; (ii) Translation invariance: for f ∈ L(F ), a ∈ F , ∫F f(x+ a) dx = ∫F f(x) dx; (iii) Compatibility with multiplicative structure: for f ∈ L(F ), α ∈ F, ∫F f(αx) dx = |α| ∫F f(x) dx. Here the absolute value of α is defined by |α| = |αt(−ν(α))|X, and we have adopted the customary integral notation ∫F (f) = ∫F f(x) dx.