Thank you very much Professor Lior. Can you kindly specify which result of Sakellaridis-Venkatesh you are referring to, as I am being unable to find it?

@Fretty Actually this question somewhat arose after I had read Gross's this note. :)

Thank you Prof. Prasad. I have been trying exactly in this method, but it seems to be too brutal attack and also unable to have any result. For $GL_n$ I have been successful in this way, but not for orthogonal or symplectic. That's why I was thinking if one somehow can use Satake transform effectively.

I used an assumption of 'Gelfand pair' so that the Hecke algebra becomes commutative. I thought that the assumption might somehow help the question.

A naive comment: There might be (I guess there is) some connection of Berry conjecture with Quantum unique ergodicity for $\Gamma\backslash\mathfrak{H}$, finite volume case.