Question

It is common to construct principal series by induction from Borel subgroup. Say $H_1$ and $H_2$ are dual representations. Both are induced representation from Borel subgroups.

Is the integration $(f_1,f_2)=\int_K f_1(k)f_2(k)dk$ the only way to construct the dual between $H_1$ and $H_2$?

Answer 1

The representations are isomorphic, if they are dual and irreducible. Given unitary representation on a Hilbert space, you are asking how many scalar products are there after identifying them.

Up to scaling, there is only at most one inner product, which makes an irreducible representation unitary.

Now why to prefer an integral over $K$?

The parabolic induced are not always irreducible, but indecomposable, e.g. $Ind_P^G |\cdotp |^{\pm 1/2}$ for $GL(2)$. To get useful information, one often restricts to $K$. Note that duality tells you that one of them contains an irreducible invariant subspace ($-1/2$) and the other contains an irreducible quotient ($1/2$).

The Iwasawa decomposition $$PK =G$$ suggests that you can realize $Res_K Ind_P^G V$ as a subspace of $L^2(K) \otimes V$. So the integral over $K$ comes from the Iwasawa decomposition.

It is very common to study discrete series or super cuspidal for reductive group over non archimedean field, via their restriction to closed subgroups, which are compact modulo the centrum and maximal with this property. In Lie theory, this is the theory of weights and in the nonarchimedean case one talks about types and fundamental strata. In both cases, a finite number of $K$ reps inside the restriction determines the representation uniquely.

There is a slightly alternative way of constructing discrete series via index theorem: Atiyah, Michael; Schmid, Wilfried A geometric construction of the discrete series for semisimple Lie groups. Invent. Math. 42 (1977), 1–62. (Reviewer: P. C. Trombi), 22E45

But I can not tell you, to what extent the constructions differ.

Answer 2

Perhaps some unasked or restated questions surrounding the literal question are as significant as the thing itself.

First, though perhaps some of this is merely English syntax rather than mathematical, "principal series" seems to refer to induction from a minimal parabolic _by_definition_, not "can". Induction of data on Levis of bigger parabolics have less reliable names, but/and are quotients and subrepresentations of "(minimal parabolic) principal series", by Harish-Chandra's subquotient theorem and Casselman(-Milicic) subrepresentation theorem.

In a different vein, for example, why is the dual of a (unitary? unitarizable? general?) principal series another principal series with parameters given in a straightforward fashion? Indeed, without worrying whether we take "smooth" induction or whatever, the duality can be proven by cleverly observing that the given integral does give the proper duality.

Other characterizations of these induced repns can use other arguments to characterize the dualities.

Changing the question: for unitary principal series, the (hermitian, not complex-bilinear) self-pairing is still given by integrating over $K$. For unitarizable, but not "unitary", the self-pairing is that integral but with one argument composed with one of the Weyl-group intertwinings.