On the formal arc space of a reductive monoid

The formal arc space L X of an algebraic variety X carries an important amount of information on singularities of X . Little is known about singularities of the formal arc scheme itself. According to Grinberg, Kazhdan [12] and Drinfeld [9] it is known, nevertheless, that the singularity of L X at a non-degenerate arc is finite dimensional i.e. for every non-degenerate arc x ∈L X , the formal completion of L X at x is isomorphic to Yy ×D∞ where Yy is the formal completion of a finite dimensional variety Y at some point y ∈ Y and D∞ is the infinite power of the formal disc. One can hope to define the intersection complex of L X via its local finite dimensional models and study the intersection complex as a measure of the singularity of L X . In this paper, we show that the trace of Frobenius function on the intersection complex is well defined on the space of non-degenerate arcs (to be defined in Section 1). The main result of this paper is the calculation of this function in the cases where X is a toric variety or a special but important class of reductive monoids. The main motivation behind this calculation is an expectation that, at least when X is an affine spherical variety under the action of a reductive group G , this function is, in a suitable sense, a generating series for an unramified local L-function (or product thereof). This expectation was stated in [19] in order to give a conceptual explanation to the Rankin-Selberg method, but the idea draws from the work of Braverman and Kazhdan who studied the Schwartz space of the basic affine space [8], and from relevant work in the geometric Langlands program [6, 5]. In the case when X is in the class of reductive monoids that we term “Lmonoids” (first introduced by Braverman and Kazhdan in [7]), a precise conjecture was formulated in [17]. It states that this function, the trace of Frobenius on the intersection complex of the formal arc space, is the generating series of the local unramified L-function for the irreducible representation of the dual group whose highest weight determines the isomorphism class of the L-monoid. This