On the Decomposition of Hecke Polynomials over Parabolic Hecke Algebras

We generalize a classical result of Andrianov on the decomposition of Hecke polynomials. Let F be a non-archimedean local fied. For every connected reductive group G, we give a criterion for when a polynomial with coefficients in the spherical parahoric Hecke algebra of G(F) decomposes over a parabolic Hecke algebra associated with a non-obtuse parabolic subgroup of G. We classify the non-obtuse parabolics. This then shows that our decomposition theorem covers all the classical cases due to Andrianov and Gritsenko. We also obtain new cases when the relative root system of G contains factors of types E6 or E7.