Abstract：

In the 1970s, Bernstein and Zelevinsky introduced a set of operators that act on the Grothendieck group of the category of admissible representations for $GL_n(Q_p)$. These operators play a crucial role in their classification of irreducible representations of $GL_n$. Later, Zelevinsky's derivatives, also known as Bernstein-Zelevinsky operators, found several important applications in automorphic theory. However, determining the Zelevinsky derivative of an irreducible representation is generally challenging. In this talk, we provide an interpretation of Zelevinsky's derivatives as dual to Lusztig's geometric inductions. As a byproduct, we derive a multiplicity formula for computing Zelevinsky's derivatives.