Star operations for affine Hecke algebras

In this paper, we consider the star operations for (graded) affine Hecke
algebras which preserve certain natural filtrations. We show that, up to inner
conjugation, there are only two such star operations for the graded Hecke
algebra: the first, denoted $\star$, corresponds to the usual star operation
from reductive $p$-adic groups, and the second, denoted $\bullet$ can be
regarded as the analogue of the compact star operation of a real group
considered by \cite{ALTV}. We explain how the star operation $\bullet$ appears
naturally in the Iwahori-spherical setting of $p$-adic groups via the
endomorphism algebras of Bernstein projectives. We also prove certain results
about the signature of $\bullet$-invariant forms and, in particular, about
$\bullet$-unitary simple modules.