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See Theorem 3, page 491 of Barut and Raczka "Theory of Group representations ..."

I am not sure, why people downvoted but I refer to a theorem of Mackey:

Let $G_1,G_2$ be two separable locally compact groups, and $H_1, H_2$ two closed subgroup, $\pi_1, \pi_2$ two unitary reps of $H_1$ resp. $H_2$, then a result of Mackey yields that

$$Ind_{H_1 \times H_2}^{G_1 \times G_2} \left( \pi_1 \otimes \pi_2 \right) \cong Ind_{H_1}^{G_1} \pi_1 \otimes Ind_{H_2}^{G_2} \pi_2.$$

Taking $\pi_j$ to be the trivial representation answer the OP's question affirmative in the most general situation.

Note that $Ind_H^G 1 = \mathcal{L}^2(H \backslash G)$ by definition, so one gets everything what one needs. The proof goes over two pages, probably it is easier in the presence of type 1 representation, since the direct integral decomposition into irreducible is not unique otherwise, but it does not seem that the reference does not assume assumes type 1. Schur's lemma is probably the key in the proof, at least it will be in the analogous statements in the compact group case.

4 added 436 characters in body

See Theorem 3, page 491 of Barut and Raczka "Theory of Group representations ..."

I am not sure, why people downvoted but I refer to a theorem of Mackey:

Let $G_1,G_2$ be two separable locally compact groups, and $H_1, H_2$ two closed subgroup, $\pi_1, \pi_2$ two unitary reps of $H_1$ resp. $H_2$, then a result of Mackey yields that

$$Ind_{H_1 \times H_2}^{G_1 \times G_2} \left( \pi_1 \otimes \pi_2 \right) \cong Ind_{H_1}^{G_1} \pi_1 \otimes Ind_{H_2}^{G_2} \pi_2.$$

Taking $\pi_j$ to be the trivial representation answer the OP's question affirmative in the most general situation.

Note that $Ind_H^G 1 = \mathcal{L}^2(H \backslash G)$ by definition, so one gets everything what one needs. The proof goes over two pages, probably it is easier in the presence of type 1 assumption enters representation, since the picturedirect integral decomposition into irreducible is not unique, but it does not seem that the reference does not assume type 1. Schur's lemma is probably the key in the proof, at least it will be in the analogous statements in the compact group case.

See Theorem 3, page 491 of Barut and Raczka "Theory of Group representations ..."..."

I am not sure, why people downvoted but I refer to a theorem of Mackey:

Let $G_1,G_2$ be two separable locally compact groups, and $H_1, H_2$ two closed subgroup, $\pi_1, \pi_2$ two unitary reps of $H_1$ resp. $H_2$, then

$$Ind_{H_1 \times H_2}^{G_1 \times G_2} \pi_1 \otimes \pi_2 \cong Ind_{H_1}^{G_1} \pi_1 \otimes Ind_{H_2}^{G_2} \pi_2.$$

Taking $\pi_j$ to be the trivial representation answer the OP's question affirmative in the most general situation.

Regarding Paul Garrett's answer: No type 1 assumption enters the picture.

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