Modular representations of reductive p-adic groups and the kernel of the trace map

The trace map of a finite group gives a perfect duality between conjugacy classes and irreducible complex representations. For reductive p-adic groups, an analogous duality holds due to the classical work of Bernstein, Deligne and Kazhdan. In this paper, we study this trace map for modular representations of reductive p-adic groups in characteristic ≠ p. The failure of the duality is measured by the kernel of the trace map on the cocenter of the ``big Hecke algebra''. Our main theorem expresses this kernel in terms of corresponding kernels for finite-dimensional Hecke algebras associated with normalizers of parahoric subgroups in suitable central quotients of Levi subgroups, reducing a problem on infinite-dimensional algebras to finite-dimensional ones. We also show that projective representations provide a natural complementary subspace on the compact part of the cocenter. For affine Hecke algebras, we prove the naive-kernel conjecture over algebraically closed fields of arbitrary characteristic and for arbitrary compatible nonzero parameters, including unequal parameters and roots of unity.