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Okay, so an automorphic form $f$ on a reductive group $G/ \mathbb{Q}$ and arithmetic subgroup $\Gamma$ is a smooth function satisfying the following conditions:

(a) invariance with respect to left $\Gamma -$ translations.

(b) Right $K -$ finiteness.

(c) Annihilated with respect to a finite co-dimension ideal of the center $Z(\mathcal{U}(\mathfrak{g}_{\mathbb{C}}))$ ( of the complexified universal enveloping algebra).

(d) Growth conditions.

Now as I understand, it is clear why condition (a) is relevant. Condition (b) comes from the idea that representation of $G$ associated to $f$ is admissible.

My understanding is that condition (d) ensures that we can "extend" the automorphic form to the cusps of the symmetric space.

I would like to know: what exactly is the intuition behind condition (c)? (I hope it is not just to provide a framework which includes classical modular forms over upper half-plane and the Maass forms.)

Also any comments about "my understanding" of conditions (a), (b) and (d) are appreciated!

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# A stupid question about Automorphic forms

Okay, so an automorphic form $f$ on a reductive group $G/ \mathbb{Q}$ and arithmetic subgroup $\Gamma$ is a smooth function satisfying the following conditions:

(a) invariance with respect to left $\Gamma -$ translations.

(b) Right $K -$ finiteness.

(c) Annihilated with respect to a finite co-dimension ideal of the center $Z(\mathcal{U}(\mathfrak{g}_{\mathbb{C}}))$ ( of the complexified universal enveloping algebra).

(d) Growth conditions.

Now as I understand, it is clear why condition (a) is relevant. Condition (b) comes from the idea that representation of $G$ associated to $f$ is admissible.

My understanding is that condition (d) ensures that we can "extend" the automorphic form to the cusps of the symmetric space.

I would like to know what exactly is the intuition behind condition (c)? (I hope it is not just to provide a framework which includes classical modular forms over upper half-plane and the Maass forms.)

Also any comments about "my understanding" of conditions (a), (b) and (d) are appreciated!