Noncommutative geometry and number theory : where arithmetic by Marcolli, Matilde; Consani, Caterina (eds)

By Marcolli, Matilde; Consani, Caterina (eds)

In recent times, quantity concept and mathematics geometry were enriched via new thoughts from noncommutative geometry, operator algebras, dynamical structures, and K-Theory. This quantity collects and offers updated examine subject matters in mathematics and noncommutative geometry and ideas from physics that time to attainable new connections among the fields of quantity conception, algebraic geometry and noncommutative geometry. The articles accumulated during this quantity current new noncommutative geometry views on classical themes of quantity thought and mathematics comparable to modular kinds, classification box thought, the speculation of reductive p-adic teams, Shimura types, the neighborhood L-factors of mathematics forms. additionally they express how mathematics appears to be like evidently in noncommutative geometry and in physics, within the residues of Feynman graphs, within the houses of noncommutative tori, and within the quantum corridor influence

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Extra resources for Noncommutative geometry and number theory : where arithmetic meets geometry and physics

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An indecomposable module VΛ,t for K as above is q-pure of weight j, or simply pure, if (i) the eigenvalues of Φ in VΛ,t are q-Weil numbers, and (ii) w(Λ) = t + j. By the argument at the end of the previous subsection, changing Φ will change the eigenvalues of Λ(Φ) only by roots of unity, and hence both conditions are independent of the choice of Φ. Also, an indecomposable VΛ,t is qK -pure of weight j if and only if, for each finite extension L of K, the restriction VΛ,t |L of VΛ,t to W DL ⊆ W DK is qL -pure of (the same) weight j.

HECKE ALGEBRA OF A REDUCTIVE p-ADIC GROUP 31 This is a parametrization of IrrI (G) by the C-points in a complex affine algebraic variety (with several components). This parametrization is not quite canonical: it depends on the specific finite sequence of elementary steps (and filtrations) connecting Hi (G) to O(L T 0 /Wf ). This parametrization will assign to each ω ∈ IrrI (G) a pair (s, γ) with s ∈L T 0 , γ a Wf -conjugacy class. More generally, the conjecture leads to a parametrization of Irr(G) by the C-points in a complex affine locally algebraic variety (with countably many components).

The Iwahori ideal in H(PGL(n)) Let G = PGL(n), let T be its standard maximal torus. Let W := X∗ (T ) Wf . Then L G0 = SL(n, C) is the Langlands dual group. Its maximal torus will be denoted L T 0 . The discrete group W is an extended Coxeter group: W = s 1 , s 2 , . . , sn Z/nZ where Z/nZ permutes cyclically the generators s1 , . . , sn . We have H(W, qF ) = H(G//I). The symmetric group Wf = Sn acts on L T 0 by permuting coordinates, and we form the quotient variety L T 0 /Sn . Let i ∈ B(G) be determined by the cuspidal pair (T, 1).

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