Yes, if a Event Registrant chooses to Pay Later during their transaction a Invoice can be automatically generated for the owed balance. Within the setup of the Event after turning on the "Allow Purchaser To Pay Later" setting, a new option called, "Automatically Create Invoice if Pay Later" will appear. Turn on that setting and you will be able to set if Partial Payment are allowed, which Payment Page will be connected to the invoice and how many days from the Transaction Date will the Invoice be due.
Along with the setup to generate the invoice you will also see the "Send Invoice via Email" setting. Turning this setting to "YES" will allow the software to automatically send that invoice via email to the same Constituent who submitted the transaction. Along with turning on the "Send Invoice via Email" there are other setting you will also need to populated. Please see the list below.
Mail Merge Template: From the picklist select with Invoice Mail Merge Letter template you would like to use as the cover letter of your automatically generated Invoice Mail Merge Letter.
Save to Document: Set to "YES" if you would like a copy of that Invoice Mail Merge Letter saved onto the Documents tab of the Constituent who submit the initial Pay Later Transaction.
Document Name: If you are saving the letter as a document you will be able to enter the Name give to each generated letter.
Document Category: You will also be able to select from this picklist what Document Category values should be assigned to each generated letter.
Invoice Email Body: The Email containing the Invoice Mail Merge letter will be separate from the Transaction Thank You Email. This field will allow you to customize the body content of that separate email. The tool bar will give you options to style this content as well as insert mail merge fields to personalize the letter.
Note: for the Email Recipient to download the Invoice Mail Merge Letter this email body but contain the document URL. By default the URL should already appear within the body with the text, "Please click here for your document". However if that URL doesn't appear you can click on the Gear icon next to the Mail Merge picklist to insert in the required URL. This dialog will also allow you to customize the URL text.
If you do not want the Invoice sent out automatically after the Pay Later transaction is submitted they you will still have the option to send out the invoice manually. This can be done using the Mail Merge function on the Invoice grid. To navigate to that grid go to Workspace>Transactions & Invoices>Invoices. Offline payment for that owed balance can also be enter here on the Invoices grid. This Payment will update both the Invoice and the Event record at the same time.