It is a genius definition, and I am still understanding. But let me state my thoughts.

In the representation theory of Lusztig type, the role of the parameter q (sometimes v) are different

1. v is the nature representation of $\mathbb{C}^\times$

2. v is the degree shift for a complex

3. v is the degree shift for a graded module

One reason to consider the involution-invariant element is that such elements are ''central''. If you know some representation theory of associative algebras or category O, you will understand that a characteristic to be simple, is to be self-adjoint (isomorphic to the dual of itself). The same case for perverse sheves.

So, the process of ''involution-invariantifying'' a basis, is similarly of the process of Gram--Schmidt, but for representations/complexes, say, finding the only unknown simple/central part one by one.

But all above is only a philosophy, the proof of Kazhdan--Lusztig conjuecture is far from trivial.

Add: usually, the natural basis are not ''central'', since most of them comes from induction from other stuff easy to be understood. This is in philosophy that our field is more than one element. If you understand q^2 as the number of elements of the field, this explanation would be of more comforts.

It is a genius definition, and I am still understanding. But let me state my thoughts.

In the representation theory of Lusztig type, the role of the parameter q (sometimes v) are different

1. v is the nature representation of $\mathbb{C}^\times$

2. v is the degree shift for a complex

3. v is the degree shift for a graded module

One reason to consider the involution-invariant element is that such elements are ''central''. If you know some representation theory of associative algebras or category O, you will understand that a characteristic to be simple, is to be self-adjoint (isomorphic to the dual of itself). The same case for perverse sheves.

So, the process of ''involution-invariantifying'' a basis, is similar to the process of Gram--Schmidt, but for representations/complexes, say, finding the only unknown simple/central part one by one.

But all above is only a philosophy, the proof of Kazhdan--Lusztig conjuecture is far from trivial.

Add: usually, the natural basis are not ''central'', since most of them comes from induction from other stuff easy to be understood. This is in philosophy due to the fact that our field is more than one element. If you understand q^2 as the number of elements of the field, this explanation would be of more comforts.

By the way, there is a lot of stuff of geometric origin coincides with Hecke algebra. It seems that any algebra with a basis parameterized by Weyl group are more or less relative to Hecke algebra.

It is a genius definition, and I am still understanding. But let me state my thoughts.

In the representation theory of Lusztig type, the role of the parameter q (sometimes v) are different

1. v is the nature representation of $\mathbb{C}^\times$

2. v is the degree shift for a complex

3. v is the degree shift for a graded module

One reason to consider the involution-invariant element is that such elements are ''central''. If you know some representation theory of associative algebras or category O, you will understand that a characteristic to be simple, is to be self-adjoint (isomorphic to the dual of itself). The same case for perverse sheves.

So, the process of ''involution-invariantifying'' a basis, is similarly of the process of Gram--Schmidt, but for representations/complexes, say, finding the only unknown simple/central part one by one.

But all above is only a philosophy, the proof of Kazhdan--Lusztig conjuecture is far from trivial.

It is a genius definition, and I am still understanding. But let me state my thoughts.

In the representation theory of Lusztig type, the role of the parameter q (sometimes v) are different

1. v is the nature representation of $\mathbb{C}^\times$

2. v is the degree shift for a complex

3. v is the degree shift for a graded module

One reason to consider the involution-invariant element is that such elements are ''central''. If you know some representation theory of associative algebras or category O, you will understand that a characteristic to be simple, is to be self-adjoint (isomorphic to the dual of itself). The same case for perverse sheves.

So, the process of ''involution-invariantifying'' a basis, is similarly of the process of Gram--Schmidt, but for representations/complexes, say, finding the only unknown simple/central part one by one.

But all above is only a philosophy, the proof of Kazhdan--Lusztig conjuecture is far from trivial.

Add: usually, the natural basis are not ''central'', since most of them comes from induction from other stuff easy to be understood. This is in philosophy that our field is more than one element. If you understand q^2 as the number of elements of the field, this explanation would be of more comforts.