Yes, you can request that your reply address in exchange be shown as your e-mail alias, but be aware of the following issues and considerations.

Mailing Lists subscriptions such as Listserv
When posting to a listserv, if your "reply to" address is different from the listserv address you will have to update the listserv.
To update a mailing list:
Unsubscribe under your Central address and re-subscribe under your alias.

Email Signature
Consider including your alias in the signature file.

To request the change, you can call the Information Technology Help Desk at 713-743-1411 or send e-mail to support@uh.edu.